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data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Κύριο μενού</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">μετακίνηση στην πλαϊνή μπάρα</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">απόκρυψη</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Πλοήγηση </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%CE%A0%CF%8D%CE%BB%CE%B7:%CE%9A%CF%8D%CF%81%CE%B9%CE%B1" title="Επισκεφθείτε την αρχική σελίδα [z]" accesskey="z"><span>Κύρια πύλη</span></a></li><li id="n-Θεματικές-πύλες" 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mw-portlet-Συμμετοχή" > <div class="vector-menu-heading"> Συμμετοχή </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/%CE%92%CE%B9%CE%BA%CE%B9%CF%80%CE%B1%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1:%CE%92%CE%BF%CE%AE%CE%B8%CE%B5%CE%B9%CE%B1" title="Το μέρος για να βρείτε αυτό που ψάχνετε"><span>Βοήθεια</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/%CE%92%CE%B9%CE%BA%CE%B9%CF%80%CE%B1%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1:%CE%A0%CF%8D%CE%BB%CE%B7_%CE%9A%CE%BF%CE%B9%CE%BD%CF%8C%CF%84%CE%B7%CF%84%CE%B1%CF%82" title="Σχετικά με το εγχείρημα, τι μπορείτε να κάνετε, πού μπορείτε να βρείτε τι"><span>Πύλη Κοινότητας</span></a></li><li id="n-pump" class="mw-list-item"><a href="/wiki/%CE%92%CE%B9%CE%BA%CE%B9%CF%80%CE%B1%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1:%CE%91%CE%B3%CE%BF%CF%81%CE%AC"><span>Αγορά</span></a></li><li id="n-recentchanges" class="mw-list-item"><a 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href="/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%9F%CE%B9%CE%A3%CF%85%CE%BD%CE%B5%CE%B9%CF%83%CF%86%CE%BF%CF%81%CE%AD%CF%82%CE%9C%CE%BF%CF%85" title="Μια λίστα με τις επεξεργασίες που έγιναν από αυτή τη διεύθυνση IP [y]" accesskey="y"><span>Συνεισφορές</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%97%CE%A3%CF%85%CE%B6%CE%AE%CF%84%CE%B7%CF%83%CE%AE%CE%9C%CE%BF%CF%85" title="Συζήτηση σχετικά με τις αλλαγές που έγιναν από αυτή τη διεύθυνση IP [n]" accesskey="n"><span>Συζήτηση για αυτή την IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv 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class="vector-pinnable-header-label">Περιεχόμενα</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">μετακίνηση στην πλαϊνή μπάρα</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">απόκρυψη</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Αρχή</div> </a> </li> <li id="toc-Όροι_των_ριζών" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Όροι_των_ριζών"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Όροι των ριζών</span> </div> </a> <ul id="toc-Όροι_των_ριζών-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ιδιότητες_και_ταυτότητες" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ιδιότητες_και_ταυτότητες"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Ιδιότητες και ταυτότητες</span> </div> </a> <button aria-controls="toc-Ιδιότητες_και_ταυτότητες-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Εναλλαγή Ιδιότητες και ταυτότητες υποενότητας</span> </button> <ul id="toc-Ιδιότητες_και_ταυτότητες-sublist" class="vector-toc-list"> <li id="toc-Γενικά" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Γενικά"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Γενικά</span> </div> </a> <ul id="toc-Γενικά-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ρίζες_άρτιας_τάξης" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ρίζες_άρτιας_τάξης"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Ρίζες άρτιας τάξης</span> </div> </a> <ul id="toc-Ρίζες_άρτιας_τάξης-sublist" class="vector-toc-list"> <li id="toc-Έχουμε_2_ρίζες" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Έχουμε_2_ρίζες"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Έχουμε 2 ρίζες</span> </div> </a> <ul id="toc-Έχουμε_2_ρίζες-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Έχουμε_0_ρίζες" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Έχουμε_0_ρίζες"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Έχουμε 0 ρίζες</span> </div> </a> <ul id="toc-Έχουμε_0_ρίζες-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ρίζες_περιττής_τάξης" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ρίζες_περιττής_τάξης"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Ρίζες περιττής τάξης</span> </div> </a> <ul id="toc-Ρίζες_περιττής_τάξης-sublist" class="vector-toc-list"> <li id="toc-Έχουμε_1_ρίζα" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Έχουμε_1_ρίζα"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Έχουμε 1 ρίζα</span> </div> </a> <ul id="toc-Έχουμε_1_ρίζα-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Δείτε_επίσης" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Δείτε_επίσης"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Δείτε επίσης</span> </div> </a> <ul id="toc-Δείτε_επίσης-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Περαιτέρω_ανάγνωση" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Περαιτέρω_ανάγνωση"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Περαιτέρω ανάγνωση</span> </div> </a> <button aria-controls="toc-Περαιτέρω_ανάγνωση-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Εναλλαγή Περαιτέρω ανάγνωση υποενότητας</span> </button> <ul id="toc-Περαιτέρω_ανάγνωση-sublist" class="vector-toc-list"> <li id="toc-Ελληνικά_άρθρα" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ελληνικά_άρθρα"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Ελληνικά άρθρα</span> </div> </a> <ul id="toc-Ελληνικά_άρθρα-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ξενόγλωσσα_άρθρα" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ξενόγλωσσα_άρθρα"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Ξενόγλωσσα άρθρα</span> </div> </a> <ul id="toc-Ξενόγλωσσα_άρθρα-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Παραπομπές" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Παραπομπές"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Παραπομπές</span> </div> </a> <ul id="toc-Παραπομπές-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Βιβλιογραφία" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Βιβλιογραφία"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Βιβλιογραφία</span> </div> </a> <ul id="toc-Βιβλιογραφία-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Περιεχόμενα" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Εναλλαγή του πίνακα περιεχομένων" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Εναλλαγή του πίνακα περιεχομένων</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Νιοστή ρίζα</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Μεταβείτε σε ένα λήμμα σε άλλη γλώσσα. Διαθέσιμο σε 69 γλώσσες" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-69" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">69 γλώσσες</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Wortelgetal" title="Wortelgetal – Αφρικάανς" lang="af" hreflang="af" data-title="Wortelgetal" data-language-autonym="Afrikaans" data-language-local-name="Αφρικάανς" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D9%86%D9%88%D9%86%D9%8A" title="جذر نوني – Αραβικά" lang="ar" hreflang="ar" data-title="جذر نوني" data-language-autonym="العربية" data-language-local-name="Αραβικά" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/K%C3%B6kalt%C4%B1" title="Kökaltı – Αζερμπαϊτζανικά" lang="az" hreflang="az" data-title="Kökaltı" data-language-autonym="Azərbaycanca" data-language-local-name="Αζερμπαϊτζανικά" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Gamot_(matematika)" title="Gamot (matematika) – Central Bikol" lang="bcl" hreflang="bcl" data-title="Gamot (matematika)" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be badge-Q17437798 badge-goodarticle mw-list-item" title="καλό λήμμα"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%BE%D1%80%D0%B0%D0%BD%D1%8C_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Корань (матэматыка) – Λευκορωσικά" lang="be" hreflang="be" data-title="Корань (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="Λευκορωσικά" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D1%80%D0%B5%D0%BD%D1%83%D0%B2%D0%B0%D0%BD%D0%B5" title="Коренуване – Βουλγαρικά" lang="bg" hreflang="bg" data-title="Коренуване" data-language-autonym="Български" data-language-local-name="Βουλγαρικά" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/N-%E0%A6%A4%E0%A6%AE_%E0%A6%AE%E0%A7%82%E0%A6%B2" title="N-তম মূল – Βεγγαλικά" lang="bn" hreflang="bn" data-title="N-তম মূল" data-language-autonym="বাংলা" data-language-local-name="Βεγγαλικά" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%98%D0%B7%D0%B0%D0%B3%D1%83%D1%83%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Изагуур (математика) – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Изагуур (математика)" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Arrel_en%C3%A8sima" title="Arrel enèsima – Καταλανικά" lang="ca" hreflang="ca" data-title="Arrel enèsima" data-language-autonym="Català" data-language-local-name="Καταλανικά" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%DB%95%DA%AF%DB%8C_n%DB%95%D9%85" title="ڕەگی nەم – Κεντρικά Κουρδικά" lang="ckb" hreflang="ckb" data-title="ڕەگی nەم" data-language-autonym="کوردی" data-language-local-name="Κεντρικά Κουρδικά" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Odmocnina" title="Odmocnina – Τσεχικά" lang="cs" hreflang="cs" data-title="Odmocnina" data-language-autonym="Čeština" data-language-local-name="Τσεχικά" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%8B%D0%BC%D0%B0%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Тымар (математика) – Τσουβασικά" lang="cv" hreflang="cv" data-title="Тымар (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Τσουβασικά" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/N%27te_rod" title="N'te rod – Δανικά" lang="da" hreflang="da" data-title="N'te rod" data-language-autonym="Dansk" data-language-local-name="Δανικά" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Wurzel_(Mathematik)" title="Wurzel (Mathematik) – Γερμανικά" lang="de" hreflang="de" data-title="Wurzel (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="Γερμανικά" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Nth_root" title="Nth root – Αγγλικά" lang="en" hreflang="en" data-title="Nth root" data-language-autonym="English" data-language-local-name="Αγγλικά" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Radicaci%C3%B3n" title="Radicación – Ισπανικά" lang="es" hreflang="es" data-title="Radicación" data-language-autonym="Español" data-language-local-name="Ισπανικά" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Juur_(matemaatika)" title="Juur (matemaatika) – Εσθονικά" lang="et" hreflang="et" data-title="Juur (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Εσθονικά" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erroketa" title="Erroketa – Βασκικά" lang="eu" hreflang="eu" data-title="Erroketa" data-language-autonym="Euskara" data-language-local-name="Βασκικά" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%B4%D9%87_%D8%B9%D8%AF%D8%AF" title="ریشه عدد – Περσικά" lang="fa" hreflang="fa" data-title="ریشه عدد" data-language-autonym="فارسی" data-language-local-name="Περσικά" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Juuri_(laskutoimitus)" title="Juuri (laskutoimitus) – Φινλανδικά" lang="fi" hreflang="fi" data-title="Juuri (laskutoimitus)" data-language-autonym="Suomi" data-language-local-name="Φινλανδικά" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Racine_d%27un_nombre" title="Racine d'un nombre – Γαλλικά" lang="fr" hreflang="fr" data-title="Racine d'un nombre" data-language-autonym="Français" data-language-local-name="Γαλλικά" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ra%C3%ADz_(matem%C3%A1ticas)" title="Raíz (matemáticas) – Γαλικιανά" lang="gl" hreflang="gl" data-title="Raíz (matemáticas)" data-language-autonym="Galego" data-language-local-name="Γαλικιανά" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Noo_fraue" title="Noo fraue – Μανξ" lang="gv" hreflang="gv" data-title="Noo fraue" data-language-autonym="Gaelg" data-language-local-name="Μανξ" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%95%D7%A8%D7%A9_%D7%A9%D7%9C_%D7%9E%D7%A1%D7%A4%D7%A8" title="שורש של מספר – Εβραϊκά" lang="he" hreflang="he" data-title="שורש של מספר" data-language-autonym="עברית" data-language-local-name="Εβραϊκά" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AE%E0%A5%82%E0%A4%B2_(%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%95%E0%A4%BE)" title="मूल (संख्या का) – Χίντι" lang="hi" hreflang="hi" data-title="मूल (संख्या का)" data-language-autonym="हिन्दी" data-language-local-name="Χίντι" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Korijen_(funkcija)" title="Korijen (funkcija) – Κροατικά" lang="hr" hreflang="hr" data-title="Korijen (funkcija)" data-language-autonym="Hrvatski" data-language-local-name="Κροατικά" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gy%C3%B6kvon%C3%A1s" title="Gyökvonás – Ουγγρικά" lang="hu" hreflang="hu" data-title="Gyökvonás" data-language-autonym="Magyar" data-language-local-name="Ουγγρικά" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D6%80%D5%B4%D5%A1%D5%BF_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Արմատ (մաթեմատիկա) – Αρμενικά" lang="hy" hreflang="hy" data-title="Արմատ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Αρμενικά" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Akar_bilangan" title="Akar bilangan – Ινδονησιακά" lang="id" hreflang="id" data-title="Akar bilangan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Ινδονησιακά" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%B3tarv%C3%ADsir" title="Rótarvísir – Ισλανδικά" lang="is" hreflang="is" data-title="Rótarvísir" data-language-autonym="Íslenska" data-language-local-name="Ισλανδικά" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Radicale_(matematica)" title="Radicale (matematica) – Ιταλικά" lang="it" hreflang="it" data-title="Radicale (matematica)" data-language-autonym="Italiano" data-language-local-name="Ιταλικά" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%AA%E6%A0%B9" title="冪根 – Ιαπωνικά" lang="ja" hreflang="ja" data-title="冪根" data-language-autonym="日本語" data-language-local-name="Ιαπωνικά" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%94%E1%83%A1%E1%83%95%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ფესვი (მათემატიკა) – Γεωργιανά" lang="ka" hreflang="ka" data-title="ფესვი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="Γεωργιανά" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D2%AF%D0%B1%D1%96%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Түбір (математика) – Καζακικά" lang="kk" hreflang="kk" data-title="Түбір (математика)" data-language-autonym="Қазақша" data-language-local-name="Καζακικά" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B3%82%E0%B2%B2" title="ಮೂಲ – Κανάντα" lang="kn" hreflang="kn" data-title="ಮೂಲ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Κανάντα" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B1%B0%EB%93%AD%EC%A0%9C%EA%B3%B1%EA%B7%BC" title="거듭제곱근 – Κορεατικά" lang="ko" hreflang="ko" data-title="거듭제곱근" data-language-autonym="한국어" data-language-local-name="Κορεατικά" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%82%D0%B0%D0%BC%D1%8B%D1%80" title="Арифметикалык тамыр – Κιργιζικά" lang="ky" hreflang="ky" data-title="Арифметикалык тамыр" data-language-autonym="Кыргызча" data-language-local-name="Κιργιζικά" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Wortel_(wiskunde)" title="Wortel (wiskunde) – Λιμβουργιανά" lang="li" hreflang="li" data-title="Wortel (wiskunde)" data-language-autonym="Limburgs" data-language-local-name="Λιμβουργιανά" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/N_%C5%A1aknis" title="N šaknis – Λιθουανικά" lang="lt" hreflang="lt" data-title="N šaknis" data-language-autonym="Lietuvių" data-language-local-name="Λιθουανικά" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sakne_(matem%C4%81tika)" title="Sakne (matemātika) – Λετονικά" lang="lv" hreflang="lv" data-title="Sakne (matemātika)" data-language-autonym="Latviešu" data-language-local-name="Λετονικά" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D1%80%D0%B5%D0%BD%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Коренување – Σλαβομακεδονικά" lang="mk" hreflang="mk" data-title="Коренување" data-language-autonym="Македонски" data-language-local-name="Σλαβομακεδονικά" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/W%C3%B6rtel_(Mathematik)" title="Wörtel (Mathematik) – Κάτω Γερμανικά" lang="nds" hreflang="nds" data-title="Wörtel (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Κάτω Γερμανικά" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wortel_(wiskunde)" title="Wortel (wiskunde) – Ολλανδικά" lang="nl" hreflang="nl" data-title="Wortel (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Ολλανδικά" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/N-te-rot" title="N-te-rot – Νορβηγικά Νινόρσκ" lang="nn" hreflang="nn" data-title="N-te-rot" data-language-autonym="Norsk nynorsk" data-language-local-name="Νορβηγικά Νινόρσκ" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/N-te-rot" title="N-te-rot – Νορβηγικά Μποκμάλ" lang="nb" hreflang="nb" data-title="N-te-rot" data-language-autonym="Norsk bokmål" data-language-local-name="Νορβηγικά Μποκμάλ" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Caaroo_N" title="Caaroo N – Ορόμο" lang="om" hreflang="om" data-title="Caaroo N" data-language-autonym="Oromoo" data-language-local-name="Ορόμο" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pierwiastkowanie" title="Pierwiastkowanie – Πολωνικά" lang="pl" hreflang="pl" data-title="Pierwiastkowanie" data-language-autonym="Polski" data-language-local-name="Πολωνικά" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Radicia%C3%A7%C3%A3o" title="Radiciação – Πορτογαλικά" lang="pt" hreflang="pt" data-title="Radiciação" data-language-autonym="Português" data-language-local-name="Πορτογαλικά" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupay_saphi" title="Yupay saphi – Κέτσουα" lang="qu" hreflang="qu" data-title="Yupay saphi" data-language-autonym="Runa Simi" data-language-local-name="Κέτσουα" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Radical_(matematic%C4%83)" title="Radical (matematică) – Ρουμανικά" lang="ro" hreflang="ro" data-title="Radical (matematică)" data-language-autonym="Română" data-language-local-name="Ρουμανικά" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="καλό λήμμα"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%80%D0%B5%D0%BD%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Корень (математика) – Ρωσικά" lang="ru" hreflang="ru" data-title="Корень (математика)" data-language-autonym="Русский" data-language-local-name="Ρωσικά" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Nth_root" title="Nth root – Simple English" lang="en-simple" hreflang="en-simple" data-title="Nth root" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Odmocnina" title="Odmocnina – Σλοβακικά" lang="sk" hreflang="sk" data-title="Odmocnina" data-language-autonym="Slovenčina" data-language-local-name="Σλοβακικά" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Korenjenje" title="Korenjenje – Σλοβενικά" lang="sl" hreflang="sl" data-title="Korenjenje" data-language-autonym="Slovenščina" data-language-local-name="Σλοβενικά" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mudzi_wenhamba" title="Mudzi wenhamba – Σόνα" lang="sn" hreflang="sn" data-title="Mudzi wenhamba" data-language-autonym="ChiShona" data-language-local-name="Σόνα" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D1%80%D0%B5%D0%BD%D0%BE%D0%B2%D0%B0%D1%9A%D0%B5" title="Кореновање – Σερβικά" lang="sr" hreflang="sr" data-title="Кореновање" data-language-autonym="Српски / srpski" data-language-local-name="Σερβικά" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rot_av_tal" title="Rot av tal – Σουηδικά" lang="sv" hreflang="sv" data-title="Rot av tal" data-language-autonym="Svenska" data-language-local-name="Σουηδικά" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/N%E0%AE%86%E0%AE%AE%E0%AF%8D_%E0%AE%AA%E0%AE%9F%E0%AE%BF_%E0%AE%AE%E0%AF%82%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="Nஆம் படி மூலம் – Ταμιλικά" lang="ta" hreflang="ta" data-title="Nஆம் படி மூலம்" data-language-autonym="தமிழ்" data-language-local-name="Ταμιλικά" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B2%E0%B8%81%E0%B8%97%E0%B8%B5%E0%B9%88_n" title="รากที่ n – Ταϊλανδικά" lang="th" hreflang="th" data-title="รากที่ n" data-language-autonym="ไทย" data-language-local-name="Ταϊλανδικά" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ugat_(matematika)" title="Ugat (matematika) – Τάγκαλογκ" lang="tl" hreflang="tl" data-title="Ugat (matematika)" data-language-autonym="Tagalog" data-language-local-name="Τάγκαλογκ" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%8A%D9%89%D9%84%D8%AA%D9%89%D8%B2_(%D9%85%D8%A7%D8%AA%DB%90%D9%85%D8%A7%D8%AA%D9%89%D9%83%D8%A7)" title="يىلتىز (ماتېماتىكا) – Ουιγουρικά" lang="ug" hreflang="ug" data-title="يىلتىز (ماتېماتىكا)" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Ουιγουρικά" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D1%80%D1%96%D0%BD%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Корінь (математика) – Ουκρανικά" lang="uk" hreflang="uk" data-title="Корінь (математика)" data-language-autonym="Українська" data-language-local-name="Ουκρανικά" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%B5%D9%85" title="اصم – Ούρντου" lang="ur" hreflang="ur" data-title="اصم" data-language-autonym="اردو" data-language-local-name="Ούρντου" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz badge-Q17437798 badge-goodarticle mw-list-item" title="καλό λήμμα"><a href="https://uz.wikipedia.org/wiki/Arifmetik_ildiz" title="Arifmetik ildiz – Ουζμπεκικά" lang="uz" hreflang="uz" data-title="Arifmetik ildiz" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Ουζμπεκικά" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C4%83n_b%E1%BA%ADc_n" title="Căn bậc n – Βιετναμικά" lang="vi" hreflang="vi" data-title="Căn bậc n" data-language-autonym="Tiếng Việt" data-language-local-name="Βιετναμικά" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Gamot_(matematika)" title="Gamot (matematika) – Γουάραϊ" lang="war" hreflang="war" data-title="Gamot (matematika)" data-language-autonym="Winaray" data-language-local-name="Γουάραϊ" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%96%B9%E6%A0%B9" title="方根 – Κινεζικά Γου" lang="wuu" hreflang="wuu" data-title="方根" data-language-autonym="吴语" data-language-local-name="Κινεζικά Γου" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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data-event-name="pinnable-header.vector-appearance.pin">μετακίνηση στην πλαϊνή μπάρα</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">απόκρυψη</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Από τη Βικιπαίδεια, την ελεύθερη εγκυκλοπαίδεια</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="el" dir="ltr"><p>Στα <a href="/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά">μαθηματικά</a> η <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστή ρίζα</b> ενός πραγματικού αριθμού <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, όταν το <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> είναι <a href="/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Φυσικός αριθμός">φυσικός αριθμός</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle >1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b523fe1c29f36fef6670b3c78f20087c5dceed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle >1}"></span>, είναι ο <a href="/wiki/%CE%A0%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Πραγματικός αριθμός">πραγματικός αριθμός</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, αν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4ddddd4f3fb0ab48b072de584c4ce3fb200d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.027ex; height:2.676ex;" alt="{\displaystyle \beta ^{\nu }=\alpha }"></span>. Η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστή ρίζα του αριθμού <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> συμβολίζεται με <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5220b20fc5d504246d7ffbaebc076d1647a8ef17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"></span>, το σύμβολο <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\;\;\;\;}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\;\;\;\;}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a600c885dec9aae8356edca82f74e0aad97df1ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.517ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\;\;\;\;}}}"></span> λέγεται <i>ριζικό</i>, το <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span></i> <i>δείκτης</i> του ριζικού, ο αριθμός <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> <i>υπόρριζο</i> και γράφεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5ab19df1a0df12717bc6e6d139b340d1a1117f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.854ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta }"></span> εάν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4ddddd4f3fb0ab48b072de584c4ce3fb200d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.027ex; height:2.676ex;" alt="{\displaystyle \beta ^{\nu }=\alpha }"></span>. Αν ο δείκτης <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span></i> είναι <a href="/wiki/%CE%86%CF%81%CF%84%CE%B9%CE%BF%CE%B9_%CE%BA%CE%B1%CE%B9_%CF%80%CE%B5%CF%81%CE%B9%CF%84%CF%84%CE%BF%CE%AF_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CE%AF" title="Άρτιοι και περιττοί αριθμοί">άρτιος</a>, ή ρίζα λέγεται <i>άρτια</i> ή <i>άρτιας τάξεως</i> και εάν είναι περιττός, η ρίζα λέγεται <i>περιττή</i> ή <i>περιττής τάξεως</i>. Ένας πραγματικός αριθμός έχει 0, 1 ή 2 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστές ρίζες, το οποίο εξαρτάται από το <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> και το <a href="/wiki/%CE%A0%CF%81%CF%8C%CF%83%CE%B7%CE%BC%CE%B1" title="Πρόσημα">πρόσημο</a> του<i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span></i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Όταν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee1c73ce4eab82eba0794ecb5b252dd47cfd189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =2}"></span>, η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστή ρίζα του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> συμβολίζεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0891fe148d778115a21a659d36dc0f5aad0237b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\alpha }}}"></span> και διαβάζεται <a href="/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1" title="Τετραγωνική ρίζα"><i>τετραγωνική</i> ή <i>δευτέρα</i> ρίζα</a> του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. Όταν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff23f7aea3cc2a731be7831d061dea5df292e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =3}"></span> συμβολίζεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ef9a9513781cebdfdd75317b5574510786cbc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{\alpha }}}"></span> και διαβάζεται <i>κυβική</i> ή <i>τρίτη</i> ρίζα του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. Όταν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =4,5,6,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =4,5,6,...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0091fc61297666750dd5e1e047e88c75fc3e068e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.635ex; height:2.509ex;" alt="{\displaystyle \nu =4,5,6,...}"></span> συμβολίζεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{4}]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe5b99bb8377bb87fc7ce2f26f7c76b956fc950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{4}]{\alpha }}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{5}]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{5}]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3b729e215290fd7b95bc9a5c27091ab3d8d990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{5}]{\alpha }}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{6}]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{6}]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f722b9a2139d9a32d17ab8cc12ee76b498b4833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{6}]{\alpha }}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00f2f395950e3698a46501d1e9aae8e8defa145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.715ex; height:0.843ex;" alt="{\displaystyle ...}"></span> και διαβάζεται <i>τέταρτη</i>, <i>πέμπτη</i>, <i>έκτη</i>, ... ρίζα του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Όροι_των_ριζών"><span id=".CE.8C.CF.81.CE.BF.CE.B9_.CF.84.CF.89.CE.BD_.CF.81.CE.B9.CE.B6.CF.8E.CE.BD"></span>Όροι των ριζών</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=1" title="Επεξεργασία ενότητας: Όροι των ριζών" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=1" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Όροι των ριζών"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Σε μία <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστή ρίζα που γράφεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5220b20fc5d504246d7ffbaebc076d1647a8ef17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"></span> έχουμε: </p> <ul><li>Τον πραγματικό αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> πού λέγεται υπόρριζο (radicand).<i><sup id="cite_ref-:2_2-0" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></i></li></ul> <ul><li>Το σύμβολο <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\;\;\;\;}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\;\;\;\;}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a600c885dec9aae8356edca82f74e0aad97df1ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.517ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\;\;\;\;}}}"></span>, που λέγεται <i>ριζικό<sup id="cite_ref-:2_2-1" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></i> (radical sign) και η οριζόντια γραμμή του πρέπει να καλύπτει πλήρως το υπόρριζο. Παράδειγμα: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\alpha \gamma }}\neq {\sqrt {\alpha }}\gamma =\gamma {\sqrt {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> <mi>γ</mi> </msqrt> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> <mi>γ</mi> <mo>=</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\alpha \gamma }}\neq {\sqrt {\alpha }}\gamma =\gamma {\sqrt {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25e9a48c05e26b6bfe0c5e53aacde0e7c21ca670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.254ex; height:3.176ex;" alt="{\displaystyle {\sqrt {\alpha \gamma }}\neq {\sqrt {\alpha }}\gamma =\gamma {\sqrt {\alpha }}}"></span></li> <li>Τον φυσικό αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>, που λέγεται δείκτης του ριζικού<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> ή βαθμός της ρίζας (degree).</li> <li>Αν ο δείκτης <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> είναι άρτιος, ή ρίζα λέγεται <i>άρτια</i> ή <i>άρτιας τάξεως</i> και εάν είναι περιττός, η ρίζα λέγεται <i>περιττή</i> ή <i>περιττής τάξεως.</i><sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>Αν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee1c73ce4eab82eba0794ecb5b252dd47cfd189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =2}"></span> συμβολίζεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0891fe148d778115a21a659d36dc0f5aad0237b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\alpha }}}"></span> και διαβάζεται <i>τετραγωνική</i> ή <i>δευτέρα</i> ρίζα του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> (square root).<sup id="cite_ref-:2_2-2" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Συνηθίζεται να μην γράφεται ο δείκτης 2 του ριζικού (δηλ. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{2}]{\;}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{2}]{\;}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1178870126a3bd118f0d40858a794176f3ec7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:2.581ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{2}]{\;}}}"></span>).</li> <li>Αν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff23f7aea3cc2a731be7831d061dea5df292e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =3}"></span> συμβολίζεται <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ef9a9513781cebdfdd75317b5574510786cbc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{\alpha }}}"></span> και διαβάζεται <i>κυβική</i> ή <i>τρίτη</i> ρίζα του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>(cube root).<sup id="cite_ref-:2_2-3" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>Δύο ρίζες λέγονται <i>ισοδύναμες</i> ή <i>ισοβάθμιοι</i>, όταν οι δείκτες του ριζικού τους ή αλλιώς οι βαθμοί τους είναι ίσοι. Για παράδειγμα οι ρίζες <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }},\;{\sqrt[{\mu }]{\beta }},\;{\sqrt[{\rho }]{\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mroot> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }},\;{\sqrt[{\mu }]{\beta }},\;{\sqrt[{\rho }]{\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11855a10743a7091e4c628001cc99c0f84b4477f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.636ex; height:3.509ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }},\;{\sqrt[{\mu }]{\beta }},\;{\sqrt[{\rho }]{\gamma }}}"></span> λέγονται ισοδύναμες όταν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\mu =\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mi>μ</mi> <mo>=</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =\mu =\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c66df527ddbceaf8f539aaf48f4e48a4c5d0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.033ex; height:2.176ex;" alt="{\displaystyle \nu =\mu =\rho }"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li><i>Συντελεστής</i> μιάς ρίζας είναι οι παράγοντες που βρίσκονται εκτός του ριζικού.<sup id="cite_ref-:6_5-0" class="reference"><a href="#cite_note-:6-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Παράδειγμα: στην ισότητα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\alpha }}\beta \gamma =\beta \gamma {\sqrt {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> <mi>β</mi> <mi>γ</mi> <mo>=</mo> <mi>β</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\alpha }}\beta \gamma =\beta \gamma {\sqrt {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd591a7383aa8cf42e03720fc6e4d34db073002b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.134ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\alpha }}\beta \gamma =\beta \gamma {\sqrt {\alpha }}}"></span>, το <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d06582df10a62dbf37c93d159f61ed142c9782b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.594ex; height:2.676ex;" alt="{\displaystyle \beta \gamma }"></span>, είναι ο συντελεστής της ρίζας.</li> <li><i>Όμοιες</i> λέγονται δύο ρίζες που έχουν τον ίδιο δείκτη (ή βαθμό) και το ίδιο υπόρριζο.<sup id="cite_ref-:6_5-1" class="reference"><a href="#cite_note-:6-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ιδιότητες_και_ταυτότητες"><span id=".CE.99.CE.B4.CE.B9.CF.8C.CF.84.CE.B7.CF.84.CE.B5.CF.82_.CE.BA.CE.B1.CE.B9_.CF.84.CE.B1.CF.85.CF.84.CF.8C.CF.84.CE.B7.CF.84.CE.B5.CF.82"></span>Ιδιότητες και ταυτότητες</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=2" title="Επεξεργασία ενότητας: Ιδιότητες και ταυτότητες" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=2" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ιδιότητες και ταυτότητες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Λόγω του ότι ο ορισμός της <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστής ρίζας βασίζεται στην εξίσωση <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4ddddd4f3fb0ab48b072de584c4ce3fb200d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.027ex; height:2.676ex;" alt="{\displaystyle \beta ^{\nu }=\alpha }"></span> , οι ιδιότητες των ριζών απορρέουν από τις ιδιότητες των <a href="/wiki/%CE%94%CF%8D%CE%BD%CE%B1%CE%BC%CE%B7_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Δύναμη (μαθηματικά)">δυνάμεων</a>.<sup id="cite_ref-:3_6-0" class="reference"><a href="#cite_note-:3-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Γενικά"><span id=".CE.93.CE.B5.CE.BD.CE.B9.CE.BA.CE.AC"></span>Γενικά</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=3" title="Επεξεργασία ενότητας: Γενικά" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=3" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Γενικά"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Πάντα ισχύει <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta \Longleftrightarrow \beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mi>β</mi> <mo stretchy="false">⟺</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta \Longleftrightarrow \beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10ec09d68b65f5ded3e0ec2550eeff07627cafb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.487ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta \Longleftrightarrow \beta ^{\nu }=\alpha }"></span>, λόγο τού ορισμού της <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστής ρίζας.<sup id="cite_ref-:0_3-2" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt[{\nu }]{\alpha }})^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt[{\nu }]{\alpha }})^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e0852fefcba3f7103173a488fe742aa2f8fb8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.922ex; height:3.009ex;" alt="{\displaystyle ({\sqrt[{\nu }]{\alpha }})^{\nu }=\alpha }"></span>, διότι ισχύει <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta \Longleftrightarrow \beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mi>β</mi> <mo stretchy="false">⟺</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta \Longleftrightarrow \beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10ec09d68b65f5ded3e0ec2550eeff07627cafb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.487ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\beta \Longleftrightarrow \beta ^{\nu }=\alpha }"></span>, τότε <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt[{\nu }]{\alpha }})^{\nu }=\beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt[{\nu }]{\alpha }})^{\nu }=\beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b956f03adf40d24cf1c4a33c5ab06ac55cb9fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.461ex; height:3.009ex;" alt="{\displaystyle ({\sqrt[{\nu }]{\alpha }})^{\nu }=\beta ^{\nu }=\alpha }"></span><sup id="cite_ref-:0_3-3" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{0}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{0}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f90a5b9d5f547b07c9b53752cc231f65f8da142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.843ex;" alt="{\displaystyle {\sqrt[{\nu }]{0}}=0}"></span>, διότι <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\nu }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\nu }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ef250c95b0ace5a288c1ba25c3eb50c1beb84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.527ex; height:2.343ex;" alt="{\displaystyle 0^{\nu }=0}"></span> για κάθε φυσικό αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>.<sup id="cite_ref-:1_7-0" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{a^{\mu }}}={\sqrt[{\nu \cdot \rho }]{a^{\mu \cdot \rho }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>⋅</mo> <mi>ρ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>⋅</mo> <mi>ρ</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{a^{\mu }}}={\sqrt[{\nu \cdot \rho }]{a^{\mu \cdot \rho }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61605464775723a565d113338b42b20f774aac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.685ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{a^{\mu }}}={\sqrt[{\nu \cdot \rho }]{a^{\mu \cdot \rho }}}}"></span>, αν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> πραγματικός και θετικός αριθμός (δηλ. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7988141e89a37e7f4deb883dbd74d9bbd6d11317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {R} }"></span>και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e01f6a4360f062e662779cb235d41c7c68a557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.749ex; height:2.343ex;" alt="{\displaystyle \alpha \geq 0}"></span>) και οι <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ,\mu ,\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ,\mu ,\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d4d21939ac84b04523c794715873e23b6af9b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.904ex; height:2.176ex;" alt="{\displaystyle \nu ,\mu ,\rho }"></span> φυσικοί αριθμοί (δηλ. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ,\mu ,\rho \in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <mi>ρ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ,\mu ,\rho \in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b4f22694f93fd82dccbe1ebefddf8cd0ece70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.295ex; height:2.676ex;" alt="{\displaystyle \nu ,\mu ,\rho \in \mathbb {Z} }"></span>).<sup id="cite_ref-:4_8-0" class="reference"><a href="#cite_note-:4-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Οι εφαρμογές της ιδιότητας αυτής είναι οι εξής:<sup id="cite_ref-:4_8-1" class="reference"><a href="#cite_note-:4-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <ol><li>Απλοποίηση ριζών, παραδείγματα: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{6}]{\alpha ^{3}}}={\sqrt[{2\cdot 3}]{\alpha ^{3}}}={\sqrt[{}]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{6}]{\alpha ^{3}}}={\sqrt[{2\cdot 3}]{\alpha ^{3}}}={\sqrt[{}]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c171b98b64b5b121c5080412f1d758beadcd15eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.49ex; height:3.676ex;" alt="{\displaystyle {\sqrt[{6}]{\alpha ^{3}}}={\sqrt[{2\cdot 3}]{\alpha ^{3}}}={\sqrt[{}]{\alpha }}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{6}]{\alpha ^{6}\cdot \beta ^{2\cdot \lambda }}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{6}]{\alpha ^{6}\cdot \beta ^{2\cdot \lambda }}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26af7538e180e1b2211f4aa075ef87773a90d309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:12.805ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{6}]{\alpha ^{6}\cdot \beta ^{2\cdot \lambda }}}=}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{2\cdot 3}]{\alpha ^{2\cdot 3}\cdot \beta ^{2\cdot \lambda }}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{2\cdot 3}]{\alpha ^{2\cdot 3}\cdot \beta ^{2\cdot \lambda }}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ad5555747a846a5f6118213da4244ebc30b71f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:14.223ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{2\cdot 3}]{\alpha ^{2\cdot 3}\cdot \beta ^{2\cdot \lambda }}}=}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{2\cdot 3}]{(\alpha ^{3}\cdot \beta ^{\lambda })^{2}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{2\cdot 3}]{(\alpha ^{3}\cdot \beta ^{\lambda })^{2}}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42808bc71265a027298ae4bcf7479730d83995a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.527ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{2\cdot 3}]{(\alpha ^{3}\cdot \beta ^{\lambda })^{2}}}=}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{\alpha ^{3}\cdot \beta ^{\lambda }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{\alpha ^{3}\cdot \beta ^{\lambda }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/467f9e05396417b68ad654a92ec626d8deee5556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.072ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{3}]{\alpha ^{3}\cdot \beta ^{\lambda }}}}"></span></li> <li>Μετατροπή ριζών σε ισοδύναμους (ισόβαθμους), απαραίτητο όταν θέλουμε να κάνουμε πράξεις μεταξύ τους. Παράδειγμα: η ρίζα 3ου βαθμού <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{a^{2}}}={\sqrt[{3\cdot 2}]{a^{2\cdot 2}}}={\sqrt[{6}]{a^{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{a^{2}}}={\sqrt[{3\cdot 2}]{a^{2\cdot 2}}}={\sqrt[{6}]{a^{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d926e4194d27aa9dfda3b099a286dd4d8d70584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.427ex; height:3.509ex;" alt="{\displaystyle {\sqrt[{3}]{a^{2}}}={\sqrt[{3\cdot 2}]{a^{2\cdot 2}}}={\sqrt[{6}]{a^{4}}}}"></span> μετατρέπεται σε 6ου βαθμού.</li></ol></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}={\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> <mo>⋅</mo> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}={\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a78df911f14c4b0ed58fc156f943ae106dbd9ad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.498ex; height:3.509ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}={\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}}"></span> ή αλλιώς <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}={\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> <mo>⋅</mo> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}={\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2893993b68412a2cb06dbab9fb7da0dd374f3381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.498ex; height:3.509ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}={\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}}"></span><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>. Σε τέτοιες πράξεις χρειάζεται οι ρίζες να μετατρέπονται σε ισόβαθμους ή και να απλοποιούνται οπότε χρησιμοποιούμε την ιδιότητα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{a^{\mu }}}={\sqrt[{\nu \cdot \rho }]{a^{\mu \cdot \rho }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>⋅</mo> <mi>ρ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>⋅</mo> <mi>ρ</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{a^{\mu }}}={\sqrt[{\nu \cdot \rho }]{a^{\mu \cdot \rho }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61605464775723a565d113338b42b20f774aac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.685ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{a^{\mu }}}={\sqrt[{\nu \cdot \rho }]{a^{\mu \cdot \rho }}}}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt[{\nu }]{\alpha }}{\sqrt[{\nu }]{\beta }}}={\sqrt[{\nu }]{\frac {\alpha }{\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> <mroot> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mi>α</mi> <mi>β</mi> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt[{\nu }]{\alpha }}{\sqrt[{\nu }]{\beta }}}={\sqrt[{\nu }]{\frac {\alpha }{\beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0dcb68078dde2340698e8015b103e1f355c3257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.238ex; height:7.176ex;" alt="{\displaystyle {\frac {\sqrt[{\nu }]{\alpha }}{\sqrt[{\nu }]{\beta }}}={\sqrt[{\nu }]{\frac {\alpha }{\beta }}}}"></span>μοιάζει και χρησιμοποιείται όπως η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}={\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> <mo>⋅</mo> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}={\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2893993b68412a2cb06dbab9fb7da0dd374f3381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.498ex; height:3.509ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}\cdot {\sqrt[{\nu }]{\beta }}\cdot {\sqrt[{\nu }]{\gamma }}={\sqrt[{\nu }]{\alpha \cdot \beta \cdot \gamma }}}"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt[{\mu }]{\alpha }})^{\nu }={\sqrt[{\mu }]{\alpha ^{\nu }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mroot> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt[{\mu }]{\alpha }})^{\nu }={\sqrt[{\mu }]{\alpha ^{\nu }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/706af3f4be7bfde363f6218a5699f8b266f5827c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.962ex; height:3.343ex;" alt="{\displaystyle ({\sqrt[{\mu }]{\alpha }})^{\nu }={\sqrt[{\mu }]{\alpha ^{\nu }}}}"></span>, δεν ισχύει πάντα (ή ισότητα δεν είναι πάντα <i>πλήρης</i>) όταν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> άρτιοι. Παράδειγμα: δεν ισχύει πάντα η ισότητα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt {4}})^{2}={\sqrt {4^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt {4}})^{2}={\sqrt {4^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3e612c45b98977f28b1bc56b1dd2dca5dbde12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.601ex; height:3.676ex;" alt="{\displaystyle ({\sqrt {4}})^{2}={\sqrt {4^{2}}}}"></span>, διότι αν πάρουμε το πρώτο μέλος της έχουμε: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt {4}})^{2}=(\pm 2)^{2}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt {4}})^{2}=(\pm 2)^{2}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160873f01268382b86b6cf506cdec81c78ea3d49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.155ex; height:3.176ex;" alt="{\displaystyle ({\sqrt {4}})^{2}=(\pm 2)^{2}=4}"></span>, ενώ από το δεύτερο έχουμε:<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {4^{2}}}={\sqrt {16}}=\pm 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>16</mn> </msqrt> </mrow> <mo>=</mo> <mo>±</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {4^{2}}}={\sqrt {16}}=\pm 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d5e39b8ec1b7b8b0cc2bd7479df33494c3bde5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.969ex; height:3.509ex;" alt="{\displaystyle {\sqrt {4^{2}}}={\sqrt {16}}=\pm 4}"></span>. Σε αυτή την περίπτωση λαμβάνουμε υπόψη μόνο τη θετική ρίζα.<sup id="cite_ref-:5_11-0" class="reference"><a href="#cite_note-:5-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\sqrt[{\mu }]{a}}}={\sqrt[{\nu \cdot \mu }]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mroot> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>⋅</mo> <mi>μ</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\sqrt[{\mu }]{a}}}={\sqrt[{\nu \cdot \mu }]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a68910c00b83882ebc8e49ab593e4875f9a2450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.348ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{\nu }]{\sqrt[{\mu }]{a}}}={\sqrt[{\nu \cdot \mu }]{a}}}"></span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha ^{\mu }}}=\alpha ^{\tfrac {\mu }{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>μ</mi> <mi>ν</mi> </mfrac> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha ^{\mu }}}=\alpha ^{\tfrac {\mu }{\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83543fac0af5906dc44406397a59faee85b1a108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.293ex; height:4.176ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha ^{\mu }}}=\alpha ^{\tfrac {\mu }{\nu }}}"></span><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ρίζες_άρτιας_τάξης"><span id=".CE.A1.CE.AF.CE.B6.CE.B5.CF.82_.CE.AC.CF.81.CF.84.CE.B9.CE.B1.CF.82_.CF.84.CE.AC.CE.BE.CE.B7.CF.82"></span>Ρίζες άρτιας τάξης</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=4" title="Επεξεργασία ενότητας: Ρίζες άρτιας τάξης" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=4" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ρίζες άρτιας τάξης"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Έχουμε_2_ρίζες"><span id=".CE.88.CF.87.CE.BF.CF.85.CE.BC.CE.B5_2_.CF.81.CE.AF.CE.B6.CE.B5.CF.82"></span>Έχουμε 2 ρίζες</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=5" title="Επεξεργασία ενότητας: Έχουμε 2 ρίζες" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=5" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Έχουμε 2 ρίζες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Εάν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> άρτιος και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span>, τότε <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\pm \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mo>±</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}=\pm \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160e264ca2a80277f2741b87249573d7bb04dee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.662ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}=\pm \beta }"></span>, δηλαδή έχει δύο ρίζες τον πραγματικό αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> και τον <a href="/wiki/%CE%91%CE%BD%CF%84%CE%AF%CF%83%CF%84%CF%81%CE%BF%CF%86%CE%BF%CF%82" title="Αντίστροφος">αντίθετό</a> του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789e8289e8f49d257d033611fbe19356b7747bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.14ex; height:2.509ex;" alt="{\displaystyle -\beta }"></span> επειδή <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4ddddd4f3fb0ab48b072de584c4ce3fb200d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.027ex; height:2.676ex;" alt="{\displaystyle \beta ^{\nu }=\alpha }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\beta )^{\nu }=\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>β</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\beta )^{\nu }=\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174865682b40a4428dfceb2dc5f44f3a1fa62223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.639ex; height:2.843ex;" alt="{\displaystyle (-\beta )^{\nu }=\alpha }"></span>. Γιαυτό το λόγο, η ισότητα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt[{\mu }]{\alpha }})^{\nu }={\sqrt[{\mu }]{\alpha ^{\nu }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mroot> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt[{\mu }]{\alpha }})^{\nu }={\sqrt[{\mu }]{\alpha ^{\nu }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/706af3f4be7bfde363f6218a5699f8b266f5827c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.962ex; height:3.343ex;" alt="{\displaystyle ({\sqrt[{\mu }]{\alpha }})^{\nu }={\sqrt[{\mu }]{\alpha ^{\nu }}}}"></span>, δεν είναι πάντα πλήρης.<sup id="cite_ref-:5_11-1" class="reference"><a href="#cite_note-:5-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd>Παράδειγμα: η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{4}]{16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{16}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e882cfc4e653db4f9468e246605b900a409705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{4}]{16}}}"></span> έχει ρίζες τον αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> και τον αντίθετό του <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e5b5b462e546b1d3d7e5f9a23efece405b2e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}"></span>, διότι <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45e90334bf6a08e457cb1ba6f83d72d7d824a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.426ex; height:2.676ex;" alt="{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2=16}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2)^{4}=(-2)\cdot (-2)\cdot (-2)\cdot (-2)=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2)^{4}=(-2)\cdot (-2)\cdot (-2)\cdot (-2)=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7dd5bee245313144536587d936f3767d39da9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.513ex; height:3.176ex;" alt="{\displaystyle (-2)^{4}=(-2)\cdot (-2)\cdot (-2)\cdot (-2)=16}"></span></dd> <dd>Σε πρακτικές εφαρμογές δεν έχουν έννοια και οι δύο ρίζες, οπότε λαμβάνουμε υπόψη μας μία από τις δύο. Συνήθως την θετική που ονομάζεται <i>κύρια <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>-οστή ρίζα</i> (principal root).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p><br /> </p> <div class="mw-heading mw-heading4"><h4 id="Έχουμε_0_ρίζες"><span id=".CE.88.CF.87.CE.BF.CF.85.CE.BC.CE.B5_0_.CF.81.CE.AF.CE.B6.CE.B5.CF.82"></span>Έχουμε 0 ρίζες</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=6" title="Επεξεργασία ενότητας: Έχουμε 0 ρίζες" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=6" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Έχουμε 0 ρίζες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Εάν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> άρτιος και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9d48dc3d4d98b4c949bf36f18559a74bc3d87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <0}"></span>, τότε η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5220b20fc5d504246d7ffbaebc076d1647a8ef17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"></span> δεν έχει ρίζα (δεν έχει έννοια), διότι δεν υπάρχει πραγματικός αριθμός <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, τέτοιος ώστε <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91904d27ab4a479c31605b617dfcc558084b6c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.441ex; height:2.676ex;" alt="{\displaystyle \beta ^{\nu }}"></span>να έχει αποτέλεσμα το <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9d48dc3d4d98b4c949bf36f18559a74bc3d87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <0}"></span>. Όπως στο προηγούμενο παράδειγμα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4}=16>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}=16>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f6996add9d4909d24cc65d25deff635034796d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.901ex; height:2.676ex;" alt="{\displaystyle 2^{4}=16>0}"></span>και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2)^{4}=16>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2)^{4}=16>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef95eeabf7198e1effd5f06e983d8a697df2dc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.518ex; height:3.176ex;" alt="{\displaystyle (-2)^{4}=16>0}"></span>, σε καμιά περίπτωση δεν έχουμε αποτέλεσμα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64f18e6b21d084770b94060cca4db532821c5b8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.133ex; height:2.343ex;" alt="{\displaystyle -16}"></span> , ώστε η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{4}]{-16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo>−</mo> <mn>16</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{-16}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b16dc4a185ac1049e43e6ad7372e8f9686312dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.069ex; height:3.176ex;" alt="{\displaystyle {\sqrt[{4}]{-16}}}"></span> να έχει ρίζα.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Η ρίζα αυτή δεν έχει έννοια στο σύνολο τον πραγματικών αριθμών και γιαυτό λέγεται <i>φανταστική παράσταση</i> και αντιμετωπίζεται με την χρήση των <a href="/wiki/%CE%A6%CE%B1%CE%BD%CF%84%CE%B1%CF%83%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Φανταστικός αριθμός">φανταστικών αριθμών</a>. Είναι <i>σημαντικό</i> όταν χρησιμοποιούμε το σύνολο των πραγματικών αριθμών να <i>εξασφαλίζεται</i> το θετικό υπόρριζο.<sup id="cite_ref-:3_6-1" class="reference"><a href="#cite_note-:3-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ρίζες_περιττής_τάξης"><span id=".CE.A1.CE.AF.CE.B6.CE.B5.CF.82_.CF.80.CE.B5.CF.81.CE.B9.CF.84.CF.84.CE.AE.CF.82_.CF.84.CE.AC.CE.BE.CE.B7.CF.82"></span>Ρίζες περιττής τάξης</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=7" title="Επεξεργασία ενότητας: Ρίζες περιττής τάξης" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=7" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ρίζες περιττής τάξης"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Έχουμε_1_ρίζα"><span id=".CE.88.CF.87.CE.BF.CF.85.CE.BC.CE.B5_1_.CF.81.CE.AF.CE.B6.CE.B1"></span>Έχουμε 1 ρίζα</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=8" title="Επεξεργασία ενότητας: Έχουμε 1 ρίζα" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=8" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Έχουμε 1 ρίζα"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Κάθε πραγματικός αριθμός έχει μόνο μία ρίζα περιττής τάξης.<sup id="cite_ref-:1_7-1" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Συγκεκριμένα: </p> <ul><li>Εάν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> περιττός και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span>, τότε η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5220b20fc5d504246d7ffbaebc076d1647a8ef17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"></span> έχει ρίζα ένα πραγματικό αριθμό. Παράδειγμα: η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513fcbe75ae8a28fbf5b6a762df26f92fa74ff1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{8}}}"></span> έχει ρίζα τον αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> διότι <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}=2\cdot 2\cdot 2=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}=2\cdot 2\cdot 2=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f605f52d30299d8589254e58ed8dfa90c2e012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.422ex; height:2.676ex;" alt="{\displaystyle 2^{3}=2\cdot 2\cdot 2=8}"></span>.</li> <li>Εάν <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> περιττός και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9d48dc3d4d98b4c949bf36f18559a74bc3d87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <0}"></span>, τότε η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5220b20fc5d504246d7ffbaebc076d1647a8ef17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha }}}"></span> έχει ρίζα ένα πραγματικό αριθμό. Παράδειγμα: η <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{-8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo>−</mo> <mn>8</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{-8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1042604e3bf8b89ddd4771fcc19dbb36b05ce423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{-8}}}"></span> έχει ρίζα τον αριθμό <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e5b5b462e546b1d3d7e5f9a23efece405b2e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}"></span> διότι <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2^{3}=-2\cdot -2\cdot -2=-8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo>−</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2^{3}=-2\cdot -2\cdot -2=-8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2e088f14cea4f20772c112a6b30a84d886104f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.462ex; height:2.843ex;" alt="{\displaystyle -2^{3}=-2\cdot -2\cdot -2=-8}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Δείτε_επίσης"><span id=".CE.94.CE.B5.CE.AF.CF.84.CE.B5_.CE.B5.CF.80.CE.AF.CF.83.CE.B7.CF.82"></span>Δείτε επίσης</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=9" title="Επεξεργασία ενότητας: Δείτε επίσης" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=9" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Δείτε επίσης"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%CE%A1%CE%AF%CE%B6%CE%B1_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Ρίζα (μαθηματικά)">Ρίζα (μαθηματικά)</a></li> <li><a href="/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1_2" class="mw-redirect" title="Τετραγωνική ρίζα 2">Τετραγωνική ρίζα του 2</a></li> <li><a href="/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1_%CF%84%CE%BF%CF%85_5" title="Τετραγωνική ρίζα του 5">Τετραγωνική ρίζα του 5</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Περαιτέρω_ανάγνωση"><span id=".CE.A0.CE.B5.CF.81.CE.B1.CE.B9.CF.84.CE.AD.CF.81.CF.89_.CE.B1.CE.BD.CE.AC.CE.B3.CE.BD.CF.89.CF.83.CE.B7"></span>Περαιτέρω ανάγνωση</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=10" title="Επεξεργασία ενότητας: Περαιτέρω ανάγνωση" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=10" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Περαιτέρω ανάγνωση"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ελληνικά_άρθρα"><span id=".CE.95.CE.BB.CE.BB.CE.B7.CE.BD.CE.B9.CE.BA.CE.AC_.CE.AC.CF.81.CE.B8.CF.81.CE.B1"></span>Ελληνικά άρθρα</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=11" title="Επεξεργασία ενότητας: Ελληνικά άρθρα" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=11" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ελληνικά άρθρα"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation Journal"><a rel="nofollow" class="external text" href="http://www.hms.gr/apothema/?s=sa&i=3928">«Για την Α' Τάξη Άλγεβρα: Ρίζες»</a>. <i><a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82_%CE%92%CE%84" title="Ευκλείδης Β΄">Ευκλείδης Β΄</a></i> (3): 109-112. 1978<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.hms.gr/apothema/?s=sa&i=3928">http://www.hms.gr/apothema/?s=sa&i=3928</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%CE%93%CE%B9%CE%B1+%CF%84%CE%B7%CE%BD+%CE%91%27+%CE%A4%CE%AC%CE%BE%CE%B7+%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1%3A+%CE%A1%CE%AF%CE%B6%CE%B5%CF%82&rft.jtitle=%5B%5B%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82+%CE%92%CE%84%5D%5D&rft.date=1978&rft.issue=3&rft.pages=109-112&rft_id=http%3A%2F%2Fwww.hms.gr%2Fapothema%2F%3Fs%3Dsa%26i%3D3928&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Κ. Μακρής (1979). <a rel="nofollow" class="external text" href="http://www.hms.gr/apothema/?s=sa&i=4126">«Για την Α' Τάξη Άλγεβρα: Ρίζες»</a>. <i><a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82_%CE%92%CE%84" title="Ευκλείδης Β΄">Ευκλείδης Β΄</a></i> (4): 158-159<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.hms.gr/apothema/?s=sa&i=4126">http://www.hms.gr/apothema/?s=sa&i=4126</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%CE%93%CE%B9%CE%B1+%CF%84%CE%B7%CE%BD+%CE%91%27+%CE%A4%CE%AC%CE%BE%CE%B7+%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1%3A+%CE%A1%CE%AF%CE%B6%CE%B5%CF%82&rft.jtitle=%5B%5B%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82+%CE%92%CE%84%5D%5D&rft.aulast=%CE%9A.+%CE%9C%CE%B1%CE%BA%CF%81%CE%AE%CF%82&rft.au=%CE%9A.+%CE%9C%CE%B1%CE%BA%CF%81%CE%AE%CF%82&rft.date=1979&rft.issue=4&rft.pages=158-159&rft_id=http%3A%2F%2Fwww.hms.gr%2Fapothema%2F%3Fs%3Dsa%26i%3D4126&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Β. Ράλλης (1981). <a rel="nofollow" class="external text" href="http://www.hms.gr/apothema/?s=sa&i=3121">«Για την Α' Τάξη Άλγεβρα: Ρίζες πραγματικών αριθμών»</a>. <i><a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82_%CE%92%CE%84" title="Ευκλείδης Β΄">Ευκλείδης Β΄</a></i> (4): 143-149<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.hms.gr/apothema/?s=sa&i=3121">http://www.hms.gr/apothema/?s=sa&i=3121</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%CE%93%CE%B9%CE%B1+%CF%84%CE%B7%CE%BD+%CE%91%27+%CE%A4%CE%AC%CE%BE%CE%B7+%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1%3A+%CE%A1%CE%AF%CE%B6%CE%B5%CF%82+%CF%80%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8E%CE%BD+%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD&rft.jtitle=%5B%5B%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82+%CE%92%CE%84%5D%5D&rft.aulast=%CE%92.+%CE%A1%CE%AC%CE%BB%CE%BB%CE%B7%CF%82&rft.au=%CE%92.+%CE%A1%CE%AC%CE%BB%CE%BB%CE%B7%CF%82&rft.date=1981&rft.issue=4&rft.pages=143-149&rft_id=http%3A%2F%2Fwww.hms.gr%2Fapothema%2F%3Fs%3Dsa%26i%3D3121&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Θ. Μαμούρης (1986). <a rel="nofollow" class="external text" href="http://www.hms.gr/apothema/?s=sa&i=3172">«Δυνάμεις και ρίζες στο σύνολο R των πραγματικών αριθμών»</a>. <i><a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82_%CE%92%CE%84" title="Ευκλείδης Β΄">Ευκλείδης Β΄</a></i> (3): 129-133<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.hms.gr/apothema/?s=sa&i=3172">http://www.hms.gr/apothema/?s=sa&i=3172</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%CE%94%CF%85%CE%BD%CE%AC%CE%BC%CE%B5%CE%B9%CF%82+%CE%BA%CE%B1%CE%B9+%CF%81%CE%AF%CE%B6%CE%B5%CF%82+%CF%83%CF%84%CE%BF+%CF%83%CF%8D%CE%BD%CE%BF%CE%BB%CE%BF+R+%CF%84%CF%89%CE%BD+%CF%80%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8E%CE%BD+%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD&rft.jtitle=%5B%5B%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82+%CE%92%CE%84%5D%5D&rft.aulast=%CE%98.+%CE%9C%CE%B1%CE%BC%CE%BF%CF%8D%CF%81%CE%B7%CF%82&rft.au=%CE%98.+%CE%9C%CE%B1%CE%BC%CE%BF%CF%8D%CF%81%CE%B7%CF%82&rft.date=1986&rft.issue=3&rft.pages=129-133&rft_id=http%3A%2F%2Fwww.hms.gr%2Fapothema%2F%3Fs%3Dsa%26i%3D3172&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Γ. Σπηλιώτης (1986). <a rel="nofollow" class="external text" href="http://www.hms.gr/apothema/?s=sa&i=3358">«Ρίζες πραγματικών αριθμών»</a>. <i><a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82_%CE%92%CE%84" title="Ευκλείδης Β΄">Ευκλείδης Β΄</a></i> (4): 214-220<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.hms.gr/apothema/?s=sa&i=3358">http://www.hms.gr/apothema/?s=sa&i=3358</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%CE%A1%CE%AF%CE%B6%CE%B5%CF%82+%CF%80%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8E%CE%BD+%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD&rft.jtitle=%5B%5B%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82+%CE%92%CE%84%5D%5D&rft.aulast=%CE%93.+%CE%A3%CF%80%CE%B7%CE%BB%CE%B9%CF%8E%CF%84%CE%B7%CF%82&rft.au=%CE%93.+%CE%A3%CF%80%CE%B7%CE%BB%CE%B9%CF%8E%CF%84%CE%B7%CF%82&rft.date=1986&rft.issue=4&rft.pages=214-220&rft_id=http%3A%2F%2Fwww.hms.gr%2Fapothema%2F%3Fs%3Dsa%26i%3D3358&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Ωρολογά, Μαρία (Αυγούστου 2012). <a rel="nofollow" class="external text" href="http://ekthetis.gr/Ekthetis012.pdf">«Ρίζες πραγματικών αριθμών και η εξίσωση x<sup>ν</sup>=α»</a>. <i>Εκθέτης Φύλλα Μαθηματικής Παιδείας</i> (12): 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=%CE%A1%CE%AF%CE%B6%CE%B5%CF%82+%CF%80%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8E%CE%BD+%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD+%CE%BA%CE%B1%CE%B9+%CE%B7+%CE%B5%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7+x%3Csup%3E%CE%BD%3C%2Fsup%3E%3D%CE%B1&rft.jtitle=%CE%95%CE%BA%CE%B8%CE%AD%CF%84%CE%B7%CF%82+%CE%A6%CF%8D%CE%BB%CE%BB%CE%B1+%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE%CF%82+%CE%A0%CE%B1%CE%B9%CE%B4%CE%B5%CE%AF%CE%B1%CF%82&rft.aulast=%CE%A9%CF%81%CE%BF%CE%BB%CE%BF%CE%B3%CE%AC&rft.aufirst=%CE%9C%CE%B1%CF%81%CE%AF%CE%B1&rft.au=%CE%A9%CF%81%CE%BF%CE%BB%CE%BF%CE%B3%CE%AC%2C%26%2332%3B%CE%9C%CE%B1%CF%81%CE%AF%CE%B1&rft.date=%CE%91%CF%85%CE%B3%CE%BF%CF%8D%CF%83%CF%84%CE%BF%CF%85+2012&rft.issue=12&rft.pages=1&rft_id=http%3A%2F%2Fekthetis.gr%2FEkthetis012.pdf&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ξενόγλωσσα_άρθρα"><span id=".CE.9E.CE.B5.CE.BD.CF.8C.CE.B3.CE.BB.CF.89.CF.83.CF.83.CE.B1_.CE.AC.CF.81.CE.B8.CF.81.CE.B1"></span>Ξενόγλωσσα άρθρα</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=12" title="Επεξεργασία ενότητας: Ξενόγλωσσα άρθρα" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=12" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ξενόγλωσσα άρθρα"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation Journal">Laufer, Henry (Μαΐου 1963). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematics-magazine_1963-05_36_3/page/157">«Finding the N th Root of a Number by Iteration»</a>. <i>Mathematics Magazine</i> <b>36</b> (3): 157–162. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F0025570X.1963.11975421">10.1080/0025570X.1963.11975421</a></span><span class="printonly">. <a rel="nofollow" class="external free" href="https://archive.org/details/sim_mathematics-magazine_1963-05_36_3/page/157">https://archive.org/details/sim_mathematics-magazine_1963-05_36_3/page/157</a></span>.</span><span class="Z3988" 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(Μαΐου 1976). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematics-magazine_1976-05_49_3/page/140">«n th Root Groups»</a>. <i>Mathematics Magazine</i> <b>49</b> (3): 140–141. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F0025570X.1976.11976563">10.1080/0025570X.1976.11976563</a></span><span class="printonly">. <a rel="nofollow" class="external free" href="https://archive.org/details/sim_mathematics-magazine_1976-05_49_3/page/140">https://archive.org/details/sim_mathematics-magazine_1976-05_49_3/page/140</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=n+th+Root+Groups&rft.jtitle=Mathematics+Magazine&rft.aulast=Kennedy&rft.aufirst=Robert+E.&rft.au=Kennedy%2C%26%2332%3BRobert+E.&rft.au=Busby%2C+Robert+W.&rft.date=%CE%9C%CE%B1%CE%90%CE%BF%CF%85+1976&rft.volume=49&rft.issue=3&rft.pages=140%E2%80%93141&rft_id=info:doi/10.1080%2F0025570X.1976.11976563&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_mathematics-magazine_1976-05_49_3%2Fpage%2F140&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Bergen, Jeffrey (Ιουνίου 2017). «Is This the Easiest Proof That n th Roots are Always Integers or Irrational?». <i>Mathematics Magazine</i> <b>90</b> (3): 225–225. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.4169%2Fmath.mag.90.3.225">10.4169/math.mag.90.3.225</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Is+This+the+Easiest+Proof+That+n+th+Roots+are+Always+Integers+or+Irrational%3F&rft.jtitle=Mathematics+Magazine&rft.aulast=Bergen&rft.aufirst=Jeffrey&rft.au=Bergen%2C%26%2332%3BJeffrey&rft.date=%CE%99%CE%BF%CF%85%CE%BD%CE%AF%CE%BF%CF%85+2017&rft.volume=90&rft.issue=3&rft.pages=225%E2%80%93225&rft_id=info:doi/10.4169%2Fmath.mag.90.3.225&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Emch, Arnold (Ιανουαρίου 1901). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1901-01_8_1/page/10">«Two Hydraulic Methods to Extract the n th Root of any Number»</a>. <i>The American Mathematical Monthly</i> <b>8</b> (1): 10–12. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F00029890.1901.12000520">10.1080/00029890.1901.12000520</a></span><span class="printonly">. <a rel="nofollow" class="external free" href="https://archive.org/details/sim_american-mathematical-monthly_1901-01_8_1/page/10">https://archive.org/details/sim_american-mathematical-monthly_1901-01_8_1/page/10</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Two+Hydraulic+Methods+to+Extract+the+n+th+Root+of+any+Number&rft.jtitle=The+American+Mathematical+Monthly&rft.aulast=Emch&rft.aufirst=Arnold&rft.au=Emch%2C%26%2332%3BArnold&rft.date=%CE%99%CE%B1%CE%BD%CE%BF%CF%85%CE%B1%CF%81%CE%AF%CE%BF%CF%85+1901&rft.volume=8&rft.issue=1&rft.pages=10%E2%80%9312&rft_id=info:doi/10.1080%2F00029890.1901.12000520&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_american-mathematical-monthly_1901-01_8_1%2Fpage%2F10&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li> <li><span class="citation Journal">Shapiro, David (1959). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27955870">«How I came across the extraction of "Nth" roots»</a>. <i>The Mathematics Teacher</i> <b>52</b> (3): 180-183<span class="printonly">. <a rel="nofollow" class="external free" href="https://www.jstor.org/stable/27955870">https://www.jstor.org/stable/27955870</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=How+I+came+across+the+extraction+of+%22Nth%22+roots&rft.jtitle=The+Mathematics+Teacher&rft.aulast=Shapiro&rft.aufirst=David&rft.au=Shapiro%2C%26%2332%3BDavid&rft.date=1959&rft.volume=52&rft.issue=3&rft.pages=180-183&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27955870&rfr_id=info:sid/el.wikipedia.org:%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Παραπομπές"><span id=".CE.A0.CE.B1.CF.81.CE.B1.CF.80.CE.BF.CE.BC.CF.80.CE.AD.CF.82"></span>Παραπομπές</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=13" title="Επεξεργασία ενότητας: Παραπομπές" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=13" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Παραπομπές"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 170</span> </li> <li id="cite_note-:2-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:2_2-0">2,0</a></sup> <sup><a href="#cite_ref-:2_2-1">2,1</a></sup> <sup><a href="#cite_ref-:2_2-2">2,2</a></sup> <sup><a href="#cite_ref-:2_2-3">2,3</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 58</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_3-0">3,0</a></sup> <sup><a href="#cite_ref-:0_3-1">3,1</a></sup> <sup><a href="#cite_ref-:0_3-2">3,2</a></sup> <sup><a href="#cite_ref-:0_3-3">3,3</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 59</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 173-174</span> </li> <li id="cite_note-:6-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:6_5-0">5,0</a></sup> <sup><a href="#cite_ref-:6_5-1">5,1</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 179</span> </li> <li id="cite_note-:3-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:3_6-0">6,0</a></sup> <sup><a href="#cite_ref-:3_6-1">6,1</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 171</span> </li> <li id="cite_note-:1-7"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:1_7-0">7,0</a></sup> <sup><a href="#cite_ref-:1_7-1">7,1</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 61</span> </li> <li id="cite_note-:4-8"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:4_8-0">8,0</a></sup> <sup><a href="#cite_ref-:4_8-1">8,1</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 173</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 175. Υπάρχει απόδειξη.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 176-177. Υπάρχει απόδειξη.</span> </li> <li id="cite_note-:5-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:5_11-0">11,0</a></sup> <sup><a href="#cite_ref-:5_11-1">11,1</a></sup></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 177-178</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 178-179. Υπάρχει απόδειξη.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 188-189. Την ισότητα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{\nu }]{\alpha ^{\mu }}}=\alpha ^{\tfrac {\mu }{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>μ</mi> <mi>ν</mi> </mfrac> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{\nu }]{\alpha ^{\mu }}}=\alpha ^{\tfrac {\mu }{\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83543fac0af5906dc44406397a59faee85b1a108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.293ex; height:4.176ex;" alt="{\displaystyle {\sqrt[{\nu }]{\alpha ^{\mu }}}=\alpha ^{\tfrac {\mu }{\nu }}}"></span>την λαμβάνει «εξ ορισμού», δηλαδή χωρίς απόδειξη.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 60-61</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text">Τόγκας Πέτρος, σελ. 60</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Βιβλιογραφία"><span id=".CE.92.CE.B9.CE.B2.CE.BB.CE.B9.CE.BF.CE.B3.CF.81.CE.B1.CF.86.CE.AF.CE.B1"></span>Βιβλιογραφία</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&veaction=edit&section=14" title="Επεξεργασία ενότητας: Βιβλιογραφία" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1&action=edit&section=14" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Βιβλιογραφία"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Τόγκας Πέτρος Γ. (Αθήνα 1959). «Άλγεβρα και Συμπλήρωμα άλγεβρας», έκδοση 26η, τόμος Α'.</li></ul> <table class="metadata plainlinks stub" style="background: transparent;" role="presentation"><tbody><tr> <td><span typeof="mw:File"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Racine_carr%C3%A9e_bleue.svg" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/30px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/45px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/60px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></td> <td><i>Αυτό το <a href="/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά">μαθηματικό</a> λήμμα χρειάζεται <a href="/wiki/%CE%92%CE%B9%CE%BA%CE%B9%CF%80%CE%B1%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1:%CE%9B%CE%AE%CE%BC%CE%BC%CE%B1%CF%84%CE%B1_%CF%80%CF%81%CE%BF%CF%82_%CE%B5%CF%80%CE%AD%CE%BA%CF%84%CE%B1%CF%83%CE%B7" title="Βικιπαίδεια:Λήμματα προς επέκταση">επέκταση</a>. 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