Racine d'un nombre — Wikipédia

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mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Racine d'un réel</span> </button> <ul id="toc-Racine_d'un_réel-sublist" class="vector-toc-list"> <li id="toc-Racine_carrée" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racine_carrée"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Racine carrée</span> </div> </a> <ul id="toc-Racine_carrée-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Racine_cubique" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racine_cubique"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Racine cubique</span> </div> </a> <ul id="toc-Racine_cubique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Racine_n-ième_d'un_nombre_réel_positif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racine_n-ième_d'un_nombre_réel_positif"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Racine <i>n</i>-ième d'un nombre réel positif</span> </div> </a> <ul id="toc-Racine_n-ième_d'un_nombre_réel_positif-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Racine_n-ième_d'un_nombre_réel_négatif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racine_n-ième_d'un_nombre_réel_négatif"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Racine <i>n</i>-ième d'un nombre réel négatif</span> </div> </a> <ul id="toc-Racine_n-ième_d'un_nombre_réel_négatif-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Les_propriétés_des_racines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Les_propriétés_des_racines"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Les propriétés des racines</span> </div> </a> <ul id="toc-Les_propriétés_des_racines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exposant_fractionnaire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exposant_fractionnaire"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Exposant fractionnaire</span> </div> </a> <ul id="toc-Exposant_fractionnaire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fonction_racine_n-ième" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fonction_racine_n-ième"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Fonction racine <i>n</i>-ième</span> </div> </a> <ul id="toc-Fonction_racine_n-ième-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Développement_en_série_entière" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Développement_en_série_entière"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>Développement en série entière</span> </div> </a> <ul id="toc-Développement_en_série_entière-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Racines_d'un_complexe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Racines_d'un_complexe"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Racines d'un complexe</span> </div> </a> <button aria-controls="toc-Racines_d'un_complexe-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>Afficher / masquer la sous-section Racines d'un complexe</span> </button> <ul id="toc-Racines_d'un_complexe-sublist" class="vector-toc-list"> <li id="toc-Nombres_réels_positifs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nombres_réels_positifs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Nombres réels positifs</span> </div> </a> <ul id="toc-Nombres_réels_positifs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Racines_de_l'unité" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racines_de_l'unité"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Racines de l'unité</span> </div> </a> <ul id="toc-Racines_de_l'unité-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Résolution_par_radicaux" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Résolution_par_radicaux"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Résolution par radicaux</span> </div> </a> <ul id="toc-Résolution_par_radicaux-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Racine_en_typographie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Racine_en_typographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Racine en typographie</span> </div> </a> <ul id="toc-Racine_en_typographie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Voir aussi</span> </div> </a> <button aria-controls="toc-Voir_aussi-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>Afficher / masquer la sous-section Voir aussi</span> </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> </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="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Racine d'un nombre</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="Aller à un article dans une autre langue. Disponible en 69 langues." > <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 langues</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 – afrikaans" lang="af" hreflang="af" data-title="Wortelgetal" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" 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 – arabe" lang="ar" hreflang="ar" data-title="جذر نوني" data-language-autonym="العربية" data-language-local-name="arabe" 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 – azerbaïdjanais" lang="az" hreflang="az" data-title="Kökaltı" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" 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="bon article"><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) – biélorusse" lang="be" hreflang="be" data-title="Корань (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="biélorusse" 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 – bulgare" lang="bg" hreflang="bg" data-title="Коренуване" data-language-autonym="Български" data-language-local-name="bulgare" 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 – bengali" lang="bn" hreflang="bn" data-title="N-তম মূল" data-language-autonym="বাংলা" data-language-local-name="bengali" 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 – catalan" lang="ca" hreflang="ca" data-title="Arrel enèsima" data-language-autonym="Català" data-language-local-name="catalan" 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 – sorani" lang="ckb" hreflang="ckb" data-title="ڕەگی nەم" data-language-autonym="کوردی" data-language-local-name="sorani" 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 – tchèque" lang="cs" hreflang="cs" data-title="Odmocnina" data-language-autonym="Čeština" data-language-local-name="tchèque" 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) – tchouvache" lang="cv" hreflang="cv" data-title="Тымар (математика)" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" 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 – danois" lang="da" hreflang="da" data-title="N'te rod" data-language-autonym="Dansk" data-language-local-name="danois" 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) – allemand" lang="de" hreflang="de" data-title="Wurzel (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9D%CE%B9%CE%BF%CF%83%CF%84%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1" title="Νιοστή ρίζα – grec" lang="el" hreflang="el" data-title="Νιοστή ρίζα" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Nth_root" title="Nth root – anglais" lang="en" hreflang="en" data-title="Nth root" data-language-autonym="English" data-language-local-name="anglais" 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 – espagnol" lang="es" hreflang="es" data-title="Radicación" data-language-autonym="Español" data-language-local-name="espagnol" 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) – estonien" lang="et" hreflang="et" data-title="Juur (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estonien" 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 – basque" lang="eu" hreflang="eu" data-title="Erroketa" data-language-autonym="Euskara" data-language-local-name="basque" 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 – persan" lang="fa" hreflang="fa" data-title="ریشه عدد" data-language-autonym="فارسی" data-language-local-name="persan" 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) – finnois" lang="fi" hreflang="fi" data-title="Juuri (laskutoimitus)" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</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) – galicien" lang="gl" hreflang="gl" data-title="Raíz (matemáticas)" data-language-autonym="Galego" data-language-local-name="galicien" 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 – mannois" lang="gv" hreflang="gv" data-title="Noo fraue" data-language-autonym="Gaelg" data-language-local-name="mannois" 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 – hébreu" lang="he" hreflang="he" data-title="שורש של מספר" data-language-autonym="עברית" data-language-local-name="hébreu" 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) – hindi" lang="hi" hreflang="hi" data-title="मूल (संख्या का)" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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) – croate" lang="hr" hreflang="hr" data-title="Korijen (funkcija)" data-language-autonym="Hrvatski" data-language-local-name="croate" 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 – hongrois" lang="hu" hreflang="hu" data-title="Gyökvonás" data-language-autonym="Magyar" data-language-local-name="hongrois" 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) – arménien" lang="hy" hreflang="hy" data-title="Արմատ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménien" 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 – indonésien" lang="id" hreflang="id" data-title="Akar bilangan" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" 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 – islandais" lang="is" hreflang="is" data-title="Rótarvísir" data-language-autonym="Íslenska" data-language-local-name="islandais" 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) – italien" lang="it" hreflang="it" data-title="Radicale (matematica)" data-language-autonym="Italiano" data-language-local-name="italien" 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 – japonais" lang="ja" hreflang="ja" data-title="冪根" data-language-autonym="日本語" data-language-local-name="japonais" 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) – géorgien" lang="ka" hreflang="ka" data-title="ფესვი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="géorgien" 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) – kazakh" lang="kk" hreflang="kk" data-title="Түбір (математика)" data-language-autonym="Қазақша" data-language-local-name="kazakh" 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 – kannada" lang="kn" hreflang="kn" data-title="ಮೂಲ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" 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 – coréen" lang="ko" hreflang="ko" data-title="거듭제곱근" data-language-autonym="한국어" data-language-local-name="coréen" 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 – kirghize" lang="ky" hreflang="ky" data-title="Арифметикалык тамыр" data-language-autonym="Кыргызча" data-language-local-name="kirghize" 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) – limbourgeois" lang="li" hreflang="li" data-title="Wortel (wiskunde)" data-language-autonym="Limburgs" data-language-local-name="limbourgeois" 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 – lituanien" lang="lt" hreflang="lt" data-title="N šaknis" data-language-autonym="Lietuvių" data-language-local-name="lituanien" 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) – letton" lang="lv" hreflang="lv" data-title="Sakne (matemātika)" data-language-autonym="Latviešu" data-language-local-name="letton" 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 – macédonien" lang="mk" hreflang="mk" data-title="Коренување" data-language-autonym="Македонски" data-language-local-name="macédonien" 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) – bas-allemand" lang="nds" hreflang="nds" data-title="Wörtel (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="bas-allemand" 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) – néerlandais" lang="nl" hreflang="nl" data-title="Wortel (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="néerlandais" 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 – norvégien nynorsk" lang="nn" hreflang="nn" data-title="N-te-rot" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" 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 – norvégien bokmål" lang="nb" hreflang="nb" data-title="N-te-rot" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" 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 – oromo" lang="om" hreflang="om" data-title="Caaroo N" data-language-autonym="Oromoo" data-language-local-name="oromo" 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 – polonais" lang="pl" hreflang="pl" data-title="Pierwiastkowanie" data-language-autonym="Polski" data-language-local-name="polonais" 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 – portugais" lang="pt" hreflang="pt" data-title="Radiciação" data-language-autonym="Português" data-language-local-name="portugais" 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 – quechua" lang="qu" hreflang="qu" data-title="Yupay saphi" data-language-autonym="Runa Simi" data-language-local-name="quechua" 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) – roumain" lang="ro" hreflang="ro" data-title="Radical (matematică)" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="bon article"><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) – russe" lang="ru" hreflang="ru" data-title="Корень (математика)" data-language-autonym="Русский" data-language-local-name="russe" 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 – slovaque" lang="sk" hreflang="sk" data-title="Odmocnina" data-language-autonym="Slovenčina" data-language-local-name="slovaque" 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 – slovène" lang="sl" hreflang="sl" data-title="Korenjenje" data-language-autonym="Slovenščina" data-language-local-name="slovène" 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 – shona" lang="sn" hreflang="sn" data-title="Mudzi wenhamba" data-language-autonym="ChiShona" data-language-local-name="shona" 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 – serbe" lang="sr" hreflang="sr" data-title="Кореновање" data-language-autonym="Српски / srpski" data-language-local-name="serbe" 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 – suédois" lang="sv" hreflang="sv" data-title="Rot av tal" data-language-autonym="Svenska" data-language-local-name="suédois" 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 – tamoul" lang="ta" hreflang="ta" data-title="Nஆம் படி மூலம்" data-language-autonym="தமிழ்" data-language-local-name="tamoul" 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 – thaï" lang="th" hreflang="th" data-title="รากที่ n" data-language-autonym="ไทย" data-language-local-name="thaï" 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) – tagalog" lang="tl" hreflang="tl" data-title="Ugat (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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) – ouïghour" lang="ug" hreflang="ug" data-title="يىلتىز (ماتېماتىكا)" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="ouïghour" 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) – ukrainien" lang="uk" hreflang="uk" data-title="Корінь (математика)" data-language-autonym="Українська" data-language-local-name="ukrainien" 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 – ourdou" lang="ur" hreflang="ur" data-title="اصم" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz badge-Q17437798 badge-goodarticle mw-list-item" title="bon article"><a href="https://uz.wikipedia.org/wiki/Arifmetik_ildiz" title="Arifmetik ildiz – ouzbek" lang="uz" hreflang="uz" data-title="Arifmetik ildiz" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" 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 – vietnamien" lang="vi" hreflang="vi" data-title="Căn bậc n" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" 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) – waray" lang="war" hreflang="war" data-title="Gamot (matematika)" data-language-autonym="Winaray" data-language-local-name="waray" 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 – wu" lang="wuu" hreflang="wuu" data-title="方根" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%96%B9%E6%A0%B9" title="方根 – chinois" lang="zh" hreflang="zh" data-title="方根" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue 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id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</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">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Racine" class="mw-disambig" title="Racine">racine</a>. </p> </div></div> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, une <b> racine <span class="texhtml"><i>n</i></span>-ième d'un nombre</b> <span class="texhtml"><i>a</i></span> est un nombre <span class="texhtml"><i>b</i></span> tel que <span class="texhtml"><i>b<sup>n</sup> = a</i></span>, où <span class="texhtml"><i>n</i></span> est un <a href="/wiki/Entier_naturel" title="Entier naturel">entier naturel</a> non nul. </p><p>Selon que l'on travaille dans l'ensemble des <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">réels</a> positifs, l'ensemble des réels ou l'ensemble des <a href="/wiki/Nombre_complexe" title="Nombre complexe">complexes</a>, le nombre de racines <span class="texhtml"><i>n</i></span>-ièmes d'un nombre peut être 0, 1, 2 ou <span class="texhtml"><i>n</i></span>. </p><p>Pour un nombre réel <span class="texhtml"><i>a</i></span> positif, il existe un unique réel <span class="texhtml"><i>b</i></span> positif tel que <span class="texhtml"><i>b<sup>n</sup> = a</i></span>. Ce réel est appelé <b>la racine <span class="texhtml"><i>n</i></span>-ième de <span class="texhtml"><i>a</i></span></b> (ou <b>racine <i>n</i>-ième principale</b> de <span class="texhtml"><i>a</i></span>) et se note <span class="racine texhtml"><sup style="margin-right: -0.5em; vertical-align: 0.8em;"><i>n</i></sup>√<span style="border-top:1px solid; padding:0 0.1em;"><i>a</i></span></span> avec le symbole <b>radical</b> (<span class="racine texhtml">√<span style="border-top:1px solid; padding:0 0.1em;"> </span></span>) ou <span class="texhtml"><i>a</i><sup>1/<i>n</i></sup></span>. La racine la plus connue est la <a href="/wiki/Racine_carr%C3%A9e" title="Racine carrée">racine carrée</a> d'un réel. Cette définition se généralise pour <span class="texhtml"><i>a</i></span> négatif et <span class="texhtml"><i>b</i></span> négatif à condition que <span class="texhtml"><i>n</i></span> soit <a href="/wiki/Parit%C3%A9_(arithm%C3%A9tique)" title="Parité (arithmétique)">impair</a>. </p><p>Le terme de racine d'un nombre ne doit pas être confondu avec celui de racine d'un <a href="/wiki/Polyn%C3%B4me" title="Polynôme">polynôme</a> qui désigne la (ou les) valeur(s) où le polynôme s'annule. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Racine_d'un_réel"><span id="Racine_d.27un_r.C3.A9el"></span>Racine d'un réel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=1" title="Modifier la section : Racine d'un réel" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=1" title="Modifier le code source de la section : Racine d'un réel"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Racine_carrée"><span id="Racine_carr.C3.A9e"></span>Racine carrée</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=2" title="Modifier la section : Racine carrée" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=2" title="Modifier le code source de la section : Racine carrée"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Racine_carr%C3%A9e" title="Racine carrée">Racine carrée</a>.</div></div> <p>Pour tout réel <span class="texhtml"><i>r</i></span> strictement positif, l'équation <i>x</i><sup>2</sup> = <span class="texhtml"><i>r</i></span> admet deux solutions réelles opposées, et lorsque <span class="texhtml"><i>r</i></span> = 0, l'équation <i>x</i><sup>2</sup> = 0 admet comme seule solution 0. </p><p>La <a href="/wiki/Racine_carr%C3%A9e" title="Racine carrée">racine carrée</a> d'un réel positif <span class="texhtml"><i>r</i></span> est par définition l'unique solution réelle positive de l'équation <i>x</i><sup>2</sup> = <span class="texhtml"><i>r</i></span> d'inconnue <i>x</i>. </p><p>Elle est notée <span class="racine texhtml">√<span style="border-top:1px solid; padding:0 0.1em;"><i>r</i></span></span>. </p> <dl><dt>Exemples</dt> <dd> <ul><li>La <a href="/wiki/Racine_carr%C3%A9e_de_deux" title="Racine carrée de deux">racine carrée de deux</a> est <span class="racine">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> = 1,414 213 56….</li> <li>Celle de trois est <span class="racine">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span> = 1,732 050 80….</li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Racine_cubique">Racine cubique</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=3" title="Modifier la section : Racine cubique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=3" title="Modifier le code source de la section : Racine cubique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Racine_cubique" title="Racine cubique">Racine cubique</a>.</div></div> <p>La <a href="/wiki/Racine_cubique" title="Racine cubique">racine cubique</a> d'un réel <i>r</i> quelconque est l'unique racine réelle de l'équation </p> <dl><dd><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 x^{3}-r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4c75616aa2f31619992ccc43a9edadf143f429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.534ex; height:2.843ex;" alt="{\displaystyle x^{3}-r=0}"></span> d'inconnue <i>x</i>.</dd></dl> <p>Elle est notée <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}]{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/718be51a7b17ff5c7c730a2d3d8cf8dedd2a7cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.985ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{r}}}"></span> . </p><p><i>Exemple :</i> </p> <ul><li>On 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 {\sqrt[{3}]{-8}}=-2}"> <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> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{-8}}=-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b130ad49972cfb951cd0334b8fce072ad343026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.976ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{-8}}=-2}"></span>. En effet <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> est le seul nombre réel dont la puissance troisième est égale à <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 -8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24cf97579cf40493908c64dc45971c781c97e78" 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 -8}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Racine_n-ième_d'un_nombre_réel_positif"><span id="Racine_n-i.C3.A8me_d.27un_nombre_r.C3.A9el_positif"></span>Racine <i>n</i>-ième d'un nombre réel positif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=4" title="Modifier la section : Racine n-ième d'un nombre réel positif" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=4" title="Modifier le code source de la section : Racine n-ième d'un nombre réel positif"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour tout entier naturel non nul <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, l'<a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</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 x\mapsto x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098425d2bbb3800078c191d5872c631469a88963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.492ex; height:2.343ex;" alt="{\displaystyle x\mapsto x^{n}}"></span> est une <a href="/wiki/Bijection" title="Bijection">bijection</a> de <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 \mathbb {R} _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{+}}"></span> sur <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 \mathbb {R} _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{+}}"></span> et donc pour tout réel <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> positif, l'équation <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 x^{n}=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c14f93ae299575135505d70bc5913a1f88250a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.695ex; height:2.343ex;" alt="{\displaystyle x^{n}=r}"></span> admet une unique solution dans <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 \mathbb {R} _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{+}}"></span>. </p><p>La <b>racine énième</b> (ou <b>racine n-ième</b>) d'un réel <i>r</i> positif (<i>r</i> ≥ 0, <i>n</i> > 0) est l'unique solution réelle positive de l'équation </p> <dl><dd><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 x^{n}-r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}-r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502d248e1e071d4369049e9a41be28315b70a806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.698ex; height:2.509ex;" alt="{\displaystyle x^{n}-r=0}"></span> d'inconnue <i>x</i>.</dd></dl> <p>Elle est notée <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[{n}]{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10eb7386bd8efe4c5b5beafe05848fbd923e1413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.985ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{r}}}"></span>. </p><p>Remarquons que la racine <i>n</i>-ième de <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> est aussi l'unique <a href="/wiki/Racine_d%27un_polyn%C3%B4me" title="Racine d'un polynôme">racine</a> positive du polynôme <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 X^{n}-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab8373f7d12a2df5f887bbe07cff6c9e64215ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.104ex; height:2.509ex;" alt="{\displaystyle X^{n}-r}"></span>. </p><p>Lorsque <i>n</i> est pair, l'équation </p> <dl><dd><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 x^{n}-r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}-r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502d248e1e071d4369049e9a41be28315b70a806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.698ex; height:2.509ex;" alt="{\displaystyle x^{n}-r=0}"></span> d'inconnue <i>x</i></dd></dl> <p>possède deux solutions qui sont <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[{n}]{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10eb7386bd8efe4c5b5beafe05848fbd923e1413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.985ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{r}}}"></span> et <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[{n}]{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt[{n}]{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd931b61a9bd0b8d0a2764e888c678dedfdc23c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.793ex; height:3.009ex;" alt="{\displaystyle -{\sqrt[{n}]{r}}}"></span>. </p><p>Lorsque <i>n</i> est impair, l'équation </p> <dl><dd><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 x^{n}-r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}-r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502d248e1e071d4369049e9a41be28315b70a806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.698ex; height:2.509ex;" alt="{\displaystyle x^{n}-r=0}"></span> d'inconnue <i>x</i></dd></dl> <p>ne possède qu'une seule solution <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[{n}]{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10eb7386bd8efe4c5b5beafe05848fbd923e1413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.985ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{r}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Racine_n-ième_d'un_nombre_réel_négatif"><span id="Racine_n-i.C3.A8me_d.27un_nombre_r.C3.A9el_n.C3.A9gatif"></span>Racine <i>n</i>-ième d'un nombre réel négatif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=5" title="Modifier la section : Racine n-ième d'un nombre réel négatif" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=5" title="Modifier le code source de la section : Racine n-ième d'un nombre réel négatif"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le traitement des racines de nombres négatifs n'est pas uniforme. Par exemple, il n'existe pas de racine carrée réelle de -1 puisque pour tout réel <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></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 x^{2}+1>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188e48375babd85fad51d363473eb2e44b48e084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}+1>0}"></span>, mais la racine cubique de -27 existe et est égale à -3. </p><p>Pour tout entier naturel <a href="/wiki/Parit%C3%A9_(arithm%C3%A9tique)" title="Parité (arithmétique)">impair</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, l'application <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 x\mapsto x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098425d2bbb3800078c191d5872c631469a88963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.492ex; height:2.343ex;" alt="{\displaystyle x\mapsto x^{n}}"></span> est une bijection de <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> sur <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> donc tout nombre réel admet exactement une racine <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-ième. </p><p>Pour tout entier naturel impair <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, la racine énième (ou racine <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-ième) d'un réel <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> quelconque est l'unique solution réelle de l'équation </p> <dl><dd><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 x^{n}-r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}-r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502d248e1e071d4369049e9a41be28315b70a806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.698ex; height:2.509ex;" alt="{\displaystyle x^{n}-r=0}"></span></dd></dl> <p>d'inconnue <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. </p><p>Il s'ensuit que les racines d'ordres impairs de nombres réels négatifs sont négatives. </p><p>Remarquons que pour les entiers naturels impairs <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> et pour tout réel <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, on a </p> <dl><dd><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[{n}]{-a}}=-{\sqrt[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo>−</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{-a}}=-{\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f844766ed3a3b580508668e627e0bccb28f3faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.046ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{-a}}=-{\sqrt[{n}]{a}}}"></span>.</dd></dl> <p>Le besoin de travailler avec des racines de nombres négatifs a conduit à la mise en place des <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombres complexes</a>, mais il y a également dans le domaine des nombres complexes des restrictions pour les racines. Voir ci-dessous. </p> <div class="mw-heading mw-heading3"><h3 id="Les_propriétés_des_racines"><span id="Les_propri.C3.A9t.C3.A9s_des_racines"></span>Les propriétés des racines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=6" title="Modifier la section : Les propriétés des racines" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=6" title="Modifier le code source de la section : Les propriétés des racines"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les règles de calcul des racines découlent des propriétés des <a href="/wiki/Puissance_d%27un_nombre" title="Puissance d'un nombre">puissances</a>. </p><p>Pour les nombres strictement positifs, <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> et <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, on a les règles de calcul suivantes : </p> <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[{n}]{a}}\cdot {\sqrt[{n}]{b}}={\sqrt[{n}]{a\cdot b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a}}\cdot {\sqrt[{n}]{b}}={\sqrt[{n}]{a\cdot b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1cdc44e892d181f2e52332608a940e0d9f9a6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.719ex; height:3.343ex;" alt="{\displaystyle {\sqrt[{n}]{a}}\cdot {\sqrt[{n}]{b}}={\sqrt[{n}]{a\cdot b}}}"></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 {\sqrt[{m}]{\sqrt[{n}]{a}}}={\sqrt[{m\cdot n}]{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>n</mi> </mrow> </mroot> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>⋅</mo> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{m}]{\sqrt[{n}]{a}}}={\sqrt[{m\cdot n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ef8ace9d360ced62f31f7d66eca161594dfc83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.812ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{m}]{\sqrt[{n}]{a}}}={\sqrt[{m\cdot n}]{a}}}"></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[{n}]{a}}{\sqrt[{n}]{b}}}={\sqrt[{n}]{\frac {a}{b}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> <mroot> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}={\sqrt[{n}]{\frac {a}{b}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a051c5fb0f6593e2d1e6a84d3ab614ffbab3eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.49ex; height:6.843ex;" alt="{\displaystyle {\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}={\sqrt[{n}]{\frac {a}{b}}}}"></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 \left({\sqrt[{n}]{a}}\right)^{m}={\sqrt[{n}]{a^{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt[{n}]{a}}\right)^{m}={\sqrt[{n}]{a^{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25d3ae659b3c55c24927a3784a3204d7fe4d1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.589ex; height:3.343ex;" alt="{\displaystyle \left({\sqrt[{n}]{a}}\right)^{m}={\sqrt[{n}]{a^{m}}}}"></span></li></ul> <p>Dans le cas des nombres négatifs, ces règles de calcul ne pourront être appliquées que si <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> et <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> sont des nombres impairs. Dans le cas des nombres complexes, elles sont à éviter. </p> <div class="mw-heading mw-heading3"><h3 id="Exposant_fractionnaire">Exposant fractionnaire</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=7" title="Modifier la section : Exposant fractionnaire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=7" title="Modifier le code source de la section : Exposant fractionnaire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans l'ensemble des réels strictement positifs, le nombre qui, élevé à la puissance <span class="texhtml"><i>n</i></span>, donne <span class="texhtml"><i>a</i></span> est noté <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[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7873203eb76042fcd24056c553de8c86054a2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{a}}}"></span>. L'idée est de noter ce nombre comme une puissance de <span class="texhtml"><i>a</i></span>, quitte à prendre un exposant non entier. Il s'agissait donc de trouver un exposant <span class="texhtml"><i>p</i></span> tel que <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 \left(a^{p}\right)^{n}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a^{p}\right)^{n}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51c4eb79cfec8d8441fb85c105a996854b82ad6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.645ex; height:3.009ex;" alt="{\displaystyle \left(a^{p}\right)^{n}=a}"></span>. En utilisant des opérations connues sur des exposants entiers que l'on généraliserait à des exposants non entiers, on obtiendrait <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 a^{pn}=a^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{pn}=a^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b9a3af8b2eb157f2f8dca40c15198b08272352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.658ex; height:2.676ex;" alt="{\displaystyle a^{pn}=a^{1}}"></span>, soit <span class="texhtml"><i>pn</i></span> = 1 et <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 p={\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6c64c9bc4b5b2db9567b4ff095d7e73fb47e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:6.588ex; height:5.176ex;" alt="{\displaystyle p={\frac {1}{n}}}"></span>. </p><p>Ainsi on peut noter la racine carrée de <span class="texhtml"><i>a</i></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 {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}"></span> ou <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 a^{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab102ab17844019d8f88084d4efa76d7a1415055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.966ex; height:3.509ex;" alt="{\displaystyle a^{\frac {1}{2}}}"></span>, la racine cubique de <span class="texhtml"><i>a</i></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}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895424fc79dfd221f984d973ac95ca277bd0e60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{a}}}"></span> ou <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 a^{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b256c868bbe505f368e8c6af9c5f347a03857502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.966ex; height:3.509ex;" alt="{\displaystyle a^{\frac {1}{3}}}"></span> et la racine <i>n</i>-ième de <span class="texhtml"><i>a</i></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[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7873203eb76042fcd24056c553de8c86054a2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{a}}}"></span> ou <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 a^{\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d741d942d62af470be1751f21021e8a140f845a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:3.343ex;" alt="{\displaystyle a^{\frac {1}{n}}}"></span>. </p><p>Cette extension des valeurs possibles pour l'exposant est due au travail de <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> et <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. On peut poursuivre le travail en observant que </p> <dl><dd><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[{n}]{a^{m}}}=\left({\sqrt[{n}]{a}}\right)^{m}=\left(a^{m}\right)^{\frac {1}{n}}=\left(a^{\frac {1}{n}}\right)^{m}=a^{\frac {m}{n}}.}"> <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>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a^{m}}}=\left({\sqrt[{n}]{a}}\right)^{m}=\left(a^{m}\right)^{\frac {1}{n}}=\left(a^{\frac {1}{n}}\right)^{m}=a^{\frac {m}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1826b35decd0ce5b3d10dfe8470a5fcc65fcf3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.134ex; height:5.009ex;" alt="{\displaystyle {\sqrt[{n}]{a^{m}}}=\left({\sqrt[{n}]{a}}\right)^{m}=\left(a^{m}\right)^{\frac {1}{n}}=\left(a^{\frac {1}{n}}\right)^{m}=a^{\frac {m}{n}}.}"></span></dd></dl> <p>et vérifier que cette notation est compatible avec les <a href="/wiki/Puissance_d%27un_nombre#Opérations_algébriques_sur_les_puissances_entières" title="Puissance d'un nombre">propriétés déjà connues sur les exposants entiers</a>. </p><p>C'est chez Newton que l'on voit apparaître pour la première fois un exposant fractionnaire. Mais Newton et Leibniz ne s'arrêteront pas là et se poseront même la question de travailler sur des exposants irrationnels sans être pour autant capables de leur donner un sens. Ce n'est qu'un siècle plus tard que ces notations prendront un sens précis avec la mise en place de la <a href="/wiki/Fonction_exponentielle" title="Fonction exponentielle">fonction exponentielle</a> et la traduction : </p> <dl><dd><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 a^{\frac {1}{n}}=\exp \left({\frac {1}{n}}\ln a\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {1}{n}}=\exp \left({\frac {1}{n}}\ln a\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828ecec40be966aa3a03d8d16b7c8e7108402132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.345ex; height:6.176ex;" alt="{\displaystyle a^{\frac {1}{n}}=\exp \left({\frac {1}{n}}\ln a\right)}"></span> pour tout réel <span class="texhtml"><i>a</i></span> strictement positif.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Fonction_racine_n-ième"><span id="Fonction_racine_n-i.C3.A8me"></span>Fonction racine <i>n</i>-ième</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=8" title="Modifier la section : Fonction racine n-ième" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=8" title="Modifier le code source de la section : Fonction racine n-ième"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fichier:RacineNieme.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/RacineNieme.svg/220px-RacineNieme.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/RacineNieme.svg/330px-RacineNieme.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/RacineNieme.svg/440px-RacineNieme.svg.png 2x" data-file-width="279" data-file-height="253" /></a><figcaption>Racine carré et racine cubique comme réciproques des fonctions carré et cube.</figcaption></figure> <p>Pour tout entier naturel non nul <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, l'application <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 x\mapsto x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098425d2bbb3800078c191d5872c631469a88963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.492ex; height:2.343ex;" alt="{\displaystyle x\mapsto x^{n}}"></span> est une bijection de ℝ<sub>+</sub> sur ℝ<sub>+</sub> dont l'<a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">application réciproque</a> est la fonction racine <i>n</i>-ième. Il est donc loisible de construire sa représentation graphique, à l'aide de celle de la <a href="/wiki/Fonction_puissance" title="Fonction puissance">fonction puissance</a> par <a href="/wiki/Sym%C3%A9trie_axiale" title="Symétrie axiale">symétrie d'axe</a> la droite d'équation <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 y=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}"></span>. </p><p>On remarque que cette fonction est <a href="/wiki/Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">continue</a> sur l'<a href="/wiki/Intervalle_(math%C3%A9matiques)" title="Intervalle (mathématiques)">intervalle</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 \left[0,+\infty \right[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[0,+\infty \right[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4493cca7f0a21fe4a0d1548b97b214b7cc0fa746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle \left[0,+\infty \right[}"></span> et l'existence à l'origine d'une tangente confondue avec l'axe des <i>y</i> donc d'une <a href="/wiki/D%C3%A9rivabilit%C3%A9" title="Dérivabilité">non-dérivabilité</a> en 0 ainsi qu'une <a href="/wiki/Branche_parabolique" title="Branche parabolique">branche parabolique</a> d'axe (<i>Ox</i>). </p><p>Les formules sur la <a href="/wiki/Bijection_r%C3%A9ciproque#Dérivabilité" title="Bijection réciproque">dérivée de la réciproque</a> permettent d'établir que la fonction racine <i>n</i>-ième est dérivable sur l'intervalle <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 \left]0,+\infty \right[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>]</mo> <mrow> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left]0,+\infty \right[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010ebc9c2d0f3af5ce1c0594c44823f107968561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle \left]0,+\infty \right[}"></span> et que sa dérivée est <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 x\mapsto {\frac {\sqrt[{n}]{x}}{nx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> <mrow> <mi>n</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\frac {\sqrt[{n}]{x}}{nx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cff5bcae21c7ee8d2645aec9be2656efd15cb6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.045ex; height:5.843ex;" alt="{\displaystyle x\mapsto {\frac {\sqrt[{n}]{x}}{nx}}}"></span>, soit encore, avec l'exposant fractionnaire <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 x\mapsto {\frac {1}{n}}x^{{\frac {1}{n}}-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\frac {1}{n}}x^{{\frac {1}{n}}-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df38ab4212c5b38a3a31a1d002f8f058fec111b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.474ex; height:5.176ex;" alt="{\displaystyle x\mapsto {\frac {1}{n}}x^{{\frac {1}{n}}-1}}"></span> montrant ainsi que la formule sur la dérivée d'une fonction puissance entière se généralise à celle d'une puissance inverse. </p> <div class="mw-heading mw-heading3"><h3 id="Développement_en_série_entière"><span id="D.C3.A9veloppement_en_s.C3.A9rie_enti.C3.A8re"></span>Développement en série entière</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=9" title="Modifier la section : Développement en série entière" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=9" title="Modifier le code source de la section : Développement en série entière"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/S%C3%A9rie_enti%C3%A8re" title="Série entière">série entière</a>.</div></div><p> Le radical ou racine peut être représenté par la <a href="/wiki/S%C3%A9rie_de_Taylor" title="Série de Taylor">série de Taylor</a> au point 1, qui s'obtient à partir de la <a href="/wiki/Formule_du_bin%C3%B4me_g%C3%A9n%C3%A9ralis%C3%A9e" title="Formule du binôme généralisée">formule du binôme généralisée</a> : pour tout réel <span class="texhtml"><i>h</i></span> tel que |<span class="texhtml"><i>h</i></span>| ≤ 1,</p><center><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[{n}]{1-h}}=1-\sum _{k=1}^{\infty }a_{k}h^{k}{\text{ avec }}a_{k}={\frac {{\frac {1}{n}}\left(1-{\frac {1}{n}}\right)\left(2-{\frac {1}{n}}\right)\cdots \left(k-1-{\frac {1}{n}}\right)}{k!}}={\frac {\displaystyle \prod _{0<s<k}(sn-1)}{k!n^{k}}}\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>−</mo> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> avec </mtext> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mi>k</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>s</mi> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> <mrow> <mi>k</mi> <mo>!</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{1-h}}=1-\sum _{k=1}^{\infty }a_{k}h^{k}{\text{ avec }}a_{k}={\frac {{\frac {1}{n}}\left(1-{\frac {1}{n}}\right)\left(2-{\frac {1}{n}}\right)\cdots \left(k-1-{\frac {1}{n}}\right)}{k!}}={\frac {\displaystyle \prod _{0<s<k}(sn-1)}{k!n^{k}}}\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e1d834dc2eeb69b64f302ead626f4203efaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:90.893ex; height:9.509ex;" alt="{\displaystyle {\sqrt[{n}]{1-h}}=1-\sum _{k=1}^{\infty }a_{k}h^{k}{\text{ avec }}a_{k}={\frac {{\frac {1}{n}}\left(1-{\frac {1}{n}}\right)\left(2-{\frac {1}{n}}\right)\cdots \left(k-1-{\frac {1}{n}}\right)}{k!}}={\frac {\displaystyle \prod _{0<s<k}(sn-1)}{k!n^{k}}}\geq 0.}"></span></center><p>En effet, cette égalité, <i>a priori</i> seulement pour |<span class="texhtml"><i>h</i></span>| < 1, assure en fait la <a href="/wiki/Convergence_normale" title="Convergence normale">convergence normale</a> sur [–1, 1] puisque</p><center><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 \sum _{k=1}^{\infty }a_{k}=\sup _{N\in \mathbb {N} ^{*}}\sum _{k=1}^{N}a_{k}=\sup _{N\in \mathbb {N} ^{*},h\in [0,1[}\sum _{k=1}^{N}a_{k}h^{k}=\sup _{h\in [0,1[}\sum _{k=1}^{\infty }a_{k}h^{k}=\sup _{h\in [0,1[}\left(1-{\sqrt[{n}]{1-h}}\right)=1<+\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">[</mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>−</mo> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo><</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }a_{k}=\sup _{N\in \mathbb {N} ^{*}}\sum _{k=1}^{N}a_{k}=\sup _{N\in \mathbb {N} ^{*},h\in [0,1[}\sum _{k=1}^{N}a_{k}h^{k}=\sup _{h\in [0,1[}\sum _{k=1}^{\infty }a_{k}h^{k}=\sup _{h\in [0,1[}\left(1-{\sqrt[{n}]{1-h}}\right)=1<+\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c4c964f92f94b7aa8c5559a9c5dab9cae5cbd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:94.016ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{\infty }a_{k}=\sup _{N\in \mathbb {N} ^{*}}\sum _{k=1}^{N}a_{k}=\sup _{N\in \mathbb {N} ^{*},h\in [0,1[}\sum _{k=1}^{N}a_{k}h^{k}=\sup _{h\in [0,1[}\sum _{k=1}^{\infty }a_{k}h^{k}=\sup _{h\in [0,1[}\left(1-{\sqrt[{n}]{1-h}}\right)=1<+\infty .}"></span></center> <p>On peut remarquer (<abbr class="abbr" title="confer (reportez-vous à/comparez avec)">cf.</abbr> « <a href="/wiki/Th%C3%A9or%C3%A8me_d%27Eisenstein" title="Théorème d'Eisenstein">Théorème d'Eisenstein</a> ») que tous les <i>n</i><sup>2<i>k</i>–1</sup><i>a<sub>k</sub></i> sont entiers (dans le <a href="/wiki/Racine_carr%C3%A9e#Fonction_réelle" title="Racine carrée">cas <i>n</i> = 2</a>, ce sont les <a href="/wiki/Nombre_de_Catalan" title="Nombre de Catalan">nombres de Catalan</a> <i>C</i><sub><i>k</i>–1</sub>). </p> <div class="mw-heading mw-heading2"><h2 id="Racines_d'un_complexe"><span id="Racines_d.27un_complexe"></span>Racines d'un complexe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=10" title="Modifier la section : Racines d'un complexe" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=10" title="Modifier le code source de la section : Racines d'un complexe"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Racine_d%27un_nombre_complexe" title="Racine d'un nombre complexe">Racine d'un nombre complexe</a>.</div></div> <p>Pour tout entier naturel non nul <span class="texhtml mvar" style="font-style:italic;">n</span>, une racine <span class="texhtml mvar" style="font-style:italic;">n</span>-ième d'un nombre complexe <i>z</i> est un nombre qui, élevé à la puissance donne <i>z</i>, c'est-à-dire une solution de l'équation </p> <dl><dd><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 x^{n}=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73211cdcec6028a414ede84cdae414272ecfd6a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.735ex; height:2.343ex;" alt="{\displaystyle x^{n}=z}"></span></dd></dl> <p>d'inconnue <span class="texhtml"><i>x</i></span>. </p><p>Lorsque <i>z</i> est différent de 0, il existe <span class="texhtml mvar" style="font-style:italic;">n</span> racines <span class="texhtml"><i>n</i></span>-ièmes distinctes de <i>z</i>. En effet, les racines <i>n</i>-ièmes d'un complexe <i>z</i> non nul sont aussi les racines du polynôme <i>X<sup>n</sup> – z</i>, qui admet bien <i>n</i> solutions dans l'ensemble des nombres complexes d'après le <a href="/wiki/Th%C3%A9or%C3%A8me_de_d%27Alembert-Gauss" class="mw-redirect" title="Théorème de d'Alembert-Gauss">théorème de d'Alembert-Gauss</a>. </p><p>Toutes les racines de n'importe quel nombre, réel ou complexe, peuvent être trouvées avec un simple <a href="/wiki/Algorithmique" title="Algorithmique">algorithme</a>. Le nombre doit d'abord être écrit sous la forme <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 a{\rm {e}}^{{\rm {i}}\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>φ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\rm {e}}^{{\rm {i}}\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b00905f31b22271809879aa075870d1ec34d231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.027ex; height:2.676ex;" alt="{\displaystyle a{\rm {e}}^{{\rm {i}}\varphi }}"></span> (voir la <a href="/wiki/Formule_d%27Euler" title="Formule d'Euler">formule d'Euler</a>). Alors, toutes les racines <span class="texhtml mvar" style="font-style:italic;">n</span>-ièmes sont données par : </p> <dl><dd><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 {\rm {e}}^{{\rm {i}}({\frac {\varphi +2k\pi }{n}})}\times {\sqrt[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{{\rm {i}}({\frac {\varphi +2k\pi }{n}})}\times {\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3667c50c3d5ed0fea92a2cf9a4e445002f23c4f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.882ex; height:4.343ex;" alt="{\displaystyle {\rm {e}}^{{\rm {i}}({\frac {\varphi +2k\pi }{n}})}\times {\sqrt[{n}]{a}}}"></span></dd></dl> <p>pour <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 k=0,1,2,\ldots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,1,2,\ldots ,n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd103867d2b8407db3fa90a3b2e009370793fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.441ex; height:2.509ex;" alt="{\displaystyle k=0,1,2,\ldots ,n-1}"></span>, où <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[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7873203eb76042fcd24056c553de8c86054a2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{a}}}"></span> représente la racine <span class="texhtml mvar" style="font-style:italic;">n</span>-ième principale de <span class="texhtml mvar" style="font-style:italic;">a</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Nombres_réels_positifs"><span id="Nombres_r.C3.A9els_positifs"></span>Nombres réels positifs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=11" title="Modifier la section : Nombres réels positifs" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=11" title="Modifier le code source de la section : Nombres réels positifs"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Toutes les solutions complexes de <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 x^{n}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f72f2b21e69fe32e9d348b1768bb5bd017fa1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.876ex; height:2.343ex;" alt="{\displaystyle x^{n}=a}"></span>, autrement dit les racines <span class="texhtml mvar" style="font-style:italic;">n</span>-ièmes de <span class="texhtml mvar" style="font-style:italic;">a</span>, où <span class="texhtml mvar" style="font-style:italic;">a</span> est un nombre réel positif, sont données par l'équation simplifiée : </p> <dl><dd><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 {\rm {e}}^{2\pi {\rm {i}}{\frac {k}{n}}}\times {\sqrt[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{2\pi {\rm {i}}{\frac {k}{n}}}\times {\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550a3bb57637b0fbb781054b74bc2ff87ef1f4c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.129ex; height:4.009ex;" alt="{\displaystyle {\rm {e}}^{2\pi {\rm {i}}{\frac {k}{n}}}\times {\sqrt[{n}]{a}}}"></span></dd></dl> <p>pour <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 k=0,1,2,\ldots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,1,2,\ldots ,n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd103867d2b8407db3fa90a3b2e009370793fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.441ex; height:2.509ex;" alt="{\displaystyle k=0,1,2,\ldots ,n-1}"></span>, où <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[{n}]{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7873203eb76042fcd24056c553de8c86054a2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{a}}}"></span> représente la racine <span class="texhtml mvar" style="font-style:italic;">n</span>-ième principale de <span class="texhtml mvar" style="font-style:italic;">a</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Racines_de_l'unité"><span id="Racines_de_l.27unit.C3.A9"></span>Racines de l'unité</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=12" title="Modifier la section : Racines de l'unité" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=12" title="Modifier le code source de la section : Racines de l'unité"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Racine_de_l%27unit%C3%A9" title="Racine de l'unité">Racine de l'unité</a>.</div></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Polyn%C3%B4me_cyclotomique" title="Polynôme cyclotomique">Polynôme cyclotomique</a>.</div></div> <p>Lorsque <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 z=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}"></span>, une telle racine s'appelle une racine <span class="texhtml mvar" style="font-style:italic;">n</span>-<b>ième de l'unité</b>, et l'ensemble des racines <span class="texhtml mvar" style="font-style:italic;">n</span>-ièmes de l'unité, noté <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 {\mathcal {U}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6852d4834cc2863616d875462d0216b1bad4d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.038ex; width:2.712ex; height:2.509ex;" alt="{\displaystyle {\mathcal {U}}_{n}}"></span>, est formé des <span class="texhtml mvar" style="font-style:italic;">n</span> racines du polynôme complexe </p> <dl><dd><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 X^{n}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d752d236be0496fbc46af44103467038b7d3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.865ex; height:2.509ex;" alt="{\displaystyle X^{n}-1.}"></span></dd></dl> <p>Il s'agit d'un <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">sous-groupe</a> <a href="/wiki/Groupe_cyclique" title="Groupe cyclique">cyclique</a> du groupe multiplicatif des <a href="/wiki/Nombre_complexe" title="Nombre complexe">complexes</a> de module 1. Il est formé des éléments <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,{\rm {e}}^{{\rm {i}}{\frac {2\pi }{n}}},{\rm {e}}^{{\rm {i}}{\frac {4\pi }{n}}},\ldots ,{\rm {e}}^{{\rm {i}}{\frac {(2n-2)\pi }{n}}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,{\rm {e}}^{{\rm {i}}{\frac {2\pi }{n}}},{\rm {e}}^{{\rm {i}}{\frac {4\pi }{n}}},\ldots ,{\rm {e}}^{{\rm {i}}{\frac {(2n-2)\pi }{n}}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d88488fc92d18ca88d3d76995462a11360ddf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.249ex; height:4.176ex;" alt="{\displaystyle \{1,{\rm {e}}^{{\rm {i}}{\frac {2\pi }{n}}},{\rm {e}}^{{\rm {i}}{\frac {4\pi }{n}}},\ldots ,{\rm {e}}^{{\rm {i}}{\frac {(2n-2)\pi }{n}}}\}}"></span> </p><p>On appelle <a href="/wiki/Racine_primitive_modulo_n" title="Racine primitive modulo n">racine <span class="texhtml mvar" style="font-style:italic;">n</span>-ième primitive de l'unité</a> tout générateur du groupe cyclique <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 {\mathcal {U}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6852d4834cc2863616d875462d0216b1bad4d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.038ex; width:2.712ex; height:2.509ex;" alt="{\displaystyle {\mathcal {U}}_{n}}"></span>. Ces racines primitives sont les éléments <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 {\rm {e}}^{{\rm {i}}{\frac {2k\pi }{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{{\rm {i}}{\frac {2k\pi }{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb42ee2bc7eeec1ea62805494a67ff594b1633f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.686ex; height:3.343ex;" alt="{\displaystyle {\rm {e}}^{{\rm {i}}{\frac {2k\pi }{n}}}}"></span> où <span class="texhtml mvar" style="font-style:italic;">k</span> est <a href="/wiki/Nombre_premier" title="Nombre premier">premier</a> avec <span class="texhtml mvar" style="font-style:italic;">n</span>. Leur nombre est égal à <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 \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067864064667dd5f8b2508b9cbf983d89788629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}"></span> où <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> désigne l'<a href="/wiki/Indicatrice_d%27Euler" title="Indicatrice d'Euler">indicatrice d'Euler</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Résolution_par_radicaux"><span id="R.C3.A9solution_par_radicaux"></span>Résolution par radicaux</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=13" title="Modifier la section : Résolution par radicaux" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=13" title="Modifier le code source de la section : Résolution par radicaux"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ludovico_Ferrari" title="Ludovico Ferrari">Ludovico Ferrari</a> a démontré que les racines des <a href="/wiki/Polyn%C3%B4me" title="Polynôme">polynômes</a> du quatrième degré pouvaient, comme pour ceux du deuxième et troisième degré, être calculées par radicaux, c'est-à-dire par un nombre fini d'opérations élémentaires sur les coefficients du polynôme, comportant des <a href="/wiki/Algorithme_de_calcul_de_la_racine_n-i%C3%A8me" title="Algorithme de calcul de la racine n-ième">calculs de racines <i>n</i>-ièmes</a>. Ceci n'est plus vrai en général pour les <a href="/wiki/%C3%89quation_quintique" title="Équation quintique">équations quintiques</a> ou d'un degré supérieur, comme l'énonce le <a href="/wiki/Th%C3%A9or%C3%A8me_d%27Abel_(alg%C3%A8bre)" title="Théorème d'Abel (algèbre)">théorème d'Abel-Ruffini</a>. Par exemple, les solutions de l'équation <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 \ x^{5}=x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x^{5}=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a816236f9ce5328fa12fe8dc79c10a35e4030dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.396ex; height:2.843ex;" alt="{\displaystyle \ x^{5}=x+1}"></span> ne peuvent pas être exprimées en termes de radicaux. </p> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/%C3%89quation_quintique" title="Équation quintique">équation quintique</a>.</div></div> <p>Pour résoudre « numériquement » n'importe quelle équation du <i>n</i>-ième degré, voir l'<a href="/wiki/Algorithme_de_recherche_d%27un_z%C3%A9ro_d%27une_fonction" title="Algorithme de recherche d'un zéro d'une fonction">algorithme de recherche de racines</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Racine_en_typographie">Racine en typographie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=14" title="Modifier la section : Racine en typographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=14" title="Modifier le code source de la section : Racine en typographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Structure_of_a_root.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Structure_of_a_root.svg/220px-Structure_of_a_root.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Structure_of_a_root.svg/330px-Structure_of_a_root.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Structure_of_a_root.svg/440px-Structure_of_a_root.svg.png 2x" data-file-width="1150" data-file-height="520" /></a><figcaption> Légende : 1. Indice ; 2. Radical ; 3. Radicande</figcaption></figure> <p>En <a href="/wiki/Typographie" title="Typographie">typographie</a>, une racine est composée de trois parties : le radical, l'indice et le radicande. </p> <ul><li>Le radical est le symbole de la racine,</li> <li>l'indice est le degré de cette racine,</li> <li>enfin, le radicande est ce qu'il y a sous le radical.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=15" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=15" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Michel Serfati, <i>La révolution symbolique</i>, chap XI, L'exponentielle après Descartes.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=16" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=16" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=17" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=17" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algorithme_de_calcul_de_la_racine_n-i%C3%A8me" title="Algorithme de calcul de la racine n-ième">Algorithme de calcul de la racine n-ième</a></li> <li><a href="/wiki/Fonction_polynomiale#Racines" title="Fonction polynomiale">Racines de fonctions polynomiales</a></li> <li><a href="/wiki/Algorithme_de_recherche_d%27un_z%C3%A9ro_d%27une_fonction" title="Algorithme de recherche d'un zéro d'une fonction">Algorithme de recherche d'un zéro d'une fonction</a></li> <li><a href="/wiki/Algorithme_de_la_potence" title="Algorithme de la potence">Algorithme de la potence</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Racine_d%27un_nombre&veaction=edit&section=18" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Racine_d%27un_nombre&action=edit&section=18" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><abbr class="abbr indicateur-langue" title="Langue : allemand">(de)</abbr> Ulrich Felgner, <i>Mathematische Semesterberichte</i>,vol. 52, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1, 2005, Springer, p. 1-7, ISSN 0720-728X (Au sujet de l'origine du signe de racine)</li> <li><abbr class="abbr indicateur-langue" title="Langue : allemand">(de)</abbr> Hans Kreul et Harald Ziebarth, <i>Mathematik leicht gemacht</i>, <abbr class="abbr" title="Sixième">6<sup>e</sup></abbr> édition, 2006 Verlag Harri Deutsch. Le chapitre complet sur la racine avec des explications, des exemples et des exercices <span class="ouvrage"><span class="noarchive">« <a rel="nofollow" class="external text" href="http://www.ziebarth-net.de/1786_probe.pdf"><cite style="font-style:normal; color:var(--color-link-red, #d73333);">disponible gratuitement en ligne</cite></a> »<sup class="plainlinks">(<a rel="nofollow" class="external text" href="https://web.archive.org/web/*/http://www.ziebarth-net.de/1786_probe.pdf">Archive.org</a> • <a rel="nofollow" class="external text" href="https://archive.wikiwix.com/cache/?url=http://www.ziebarth-net.de/1786_probe.pdf">Wikiwix</a> • <a rel="nofollow" class="external text" href="https://archive.is/http://www.ziebarth-net.de/1786_probe.pdf">Archive.is</a> • <a rel="nofollow" class="external text" href="https://webcache.googleusercontent.com/search?hl=fr&q=cache:http://www.ziebarth-net.de/1786_probe.pdf">Google</a> • <a href="/wiki/Projet:Correction_des_liens_externes#J'ai_trouvé_un_lien_mort,_que_faire_?" title="Projet:Correction des liens externes">Que faire ?</a>)</sup></span></span> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-3-8171-1786-4" title="Spécial:Ouvrages de référence/978-3-8171-1786-4"><span class="nowrap">978-3-8171-1786-4</span></a>)</small></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Fonctions_math%C3%A9matiques_usuelles" title="Modèle:Palette Fonctions mathématiques usuelles"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Fonctions_math%C3%A9matiques_usuelles&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Fonction_(math%C3%A9matiques)" title="Fonction (mathématiques)">Fonctions mathématiques usuelles</a></div></th> </tr> <tr> <th class="navbox-group" style="width:200px"><a href="/wiki/Fonction_alg%C3%A9brique" title="Fonction algébrique">Fonction algébrique</a> <a href="/wiki/Fonction_rationnelle" title="Fonction rationnelle">rationnelle</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Fonction_polynomiale" title="Fonction polynomiale">Fonction polynomiale</a></li> <li><a href="/wiki/Fraction_rationnelle" title="Fraction rationnelle">Fonction fractionnaire</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:200px">Fonction algébrique irrationnelle</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Fonction_puissance" title="Fonction puissance">Fonction puissance</a> / <a class="mw-selflink selflink">Fonction racine</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:200px"><a href="/wiki/Fonction_transcendante" title="Fonction transcendante">Fonction transcendante</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Logarithme" title="Logarithme">Fonction logarithmique</a> / <a href="/wiki/Exponentielle_de_base_a" title="Exponentielle de base a">Fonction exponentielle de base a</a> <ul><li><a href="/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien">Fonction logarithme naturel</a> / <a href="/wiki/Fonction_exponentielle" title="Fonction exponentielle">Fonction exponentielle</a></li></ul></li> <li><a href="/wiki/Fonction_trigonom%C3%A9trique" title="Fonction trigonométrique">Fonction circulaire</a> / <a href="/wiki/Fonction_circulaire_r%C3%A9ciproque" title="Fonction circulaire réciproque">Fonction circulaire réciproque</a></li> <li><a href="/wiki/Fonction_hyperbolique" title="Fonction hyperbolique">Fonction hyperbolique</a> / <a href="/wiki/Fonction_hyperbolique#Applications_réciproques" title="Fonction hyperbolique">Fonction hyperbolique réciproque</a></li> <li><a href="/wiki/Fonction_elliptique" title="Fonction elliptique">Fonction elliptique</a> / <a href="/wiki/Int%C3%A9grale_elliptique" title="Intégrale elliptique">Fonction intégrale elliptique</a></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/36px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/48px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail:Mathématiques">Portail des mathématiques</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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