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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Přihlášení</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Stránky pro odhlášené editory <a href="/wiki/N%C3%A1pov%C4%9Bda:%C3%9Avod" aria-label="Více informací o editování"><span>dozvědět se více</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Moje_p%C5%99%C3%ADsp%C4%9Bvky" title="Seznam editací provedených z této IP adresy [y]" accesskey="y"><span>Příspěvky</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Moje_diskuse" title="Diskuse o editacích provedených z této IP adresy [n]" accesskey="n"><span>Diskuse</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Obsah" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(úvod)</div> </a> </li> <li id="toc-Definice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definice</span> </div> </a> <button aria-controls="toc-Definice-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>Přepnout podsekci Definice</span> </button> <ul id="toc-Definice-sublist" class="vector-toc-list"> <li id="toc-Alternativní_definice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternativní_definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Alternativní definice</span> </div> </a> <ul id="toc-Alternativní_definice-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vlastnosti</span> </div> </a> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mocniny_nuly" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mocniny_nuly"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mocniny nuly</span> </div> </a> <button aria-controls="toc-Mocniny_nuly-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>Přepnout podsekci Mocniny nuly</span> </button> <ul id="toc-Mocniny_nuly-sublist" class="vector-toc-list"> <li id="toc-Nula_na_nultou" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nula_na_nultou"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Nula na nultou</span> </div> </a> <ul id="toc-Nula_na_nultou-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Zvláštní_mocniny" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zvláštní_mocniny"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Zvláštní mocniny</span> </div> </a> <ul id="toc-Zvláštní_mocniny-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Obsah" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Obsah" > <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="Přepnout obsah" > <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">Přepnout obsah</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">Umocňování</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="Přejděte k článku v jiném jazyce. Je dostupný v 90 jazycích" > <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-90" 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">90 jazyků</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/Magsverheffing" title="Magsverheffing – afrikánština" lang="af" hreflang="af" data-title="Magsverheffing" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Potenz_(Mathematik)" title="Potenz (Mathematik) – němčina (Švýcarsko)" lang="gsw" hreflang="gsw" data-title="Potenz (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="němčina (Švýcarsko)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%95%E1%88%B4%E1%89%B5" title="ንሴት – amharština" lang="am" hreflang="am" data-title="ንሴት" data-language-autonym="አማርኛ" data-language-local-name="amharština" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%81%D8%B9_%D8%A3%D8%B3%D9%8A" title="رفع أسي – arabština" lang="ar" hreflang="ar" data-title="رفع أسي" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Potenciaci%C3%B3n" title="Potenciación – asturština" lang="ast" hreflang="ast" data-title="Potenciación" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Q%C3%BCvv%C9%99t%C9%99_y%C3%BCks%C9%99ltm%C9%99" title="Qüvvətə yüksəltmə – ázerbájdžánština" lang="az" hreflang="az" data-title="Qüvvətə yüksəltmə" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%94%D3%99%D1%80%D3%99%D0%B6%D3%99%D0%B3%D3%99_%D0%BA%D2%AF%D1%82%D3%99%D1%80%D0%B5%D2%AF" title="Дәрәжәгә күтәреү – baškirština" lang="ba" hreflang="ba" data-title="Дәрәжәгә күтәреү" data-language-autonym="Башҡортса" data-language-local-name="baškirština" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Eksponentasyon" title="Eksponentasyon – Central Bikol" lang="bcl" hreflang="bcl" data-title="Eksponentasyon" 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 mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D1%8F%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Ступеняванне – běloruština" lang="be" hreflang="be" data-title="Ступеняванне" data-language-autonym="Беларуская" data-language-local-name="běloruština" 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%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%83%D0%B2%D0%B0%D0%BD%D0%B5_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Степенуване (математика) – bulharština" lang="bg" hreflang="bg" data-title="Степенуване (математика)" data-language-autonym="Български" data-language-local-name="bulharština" 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/%E0%A6%B8%E0%A7%82%E0%A6%9A%E0%A6%95%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="সূচকীকরণ – bengálština" lang="bn" hreflang="bn" data-title="সূচকীকরণ" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Eksponent" title="Eksponent – bosenština" lang="bs" hreflang="bs" data-title="Eksponent" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%97%D1%8D%D1%80%D0%B3%D1%8D%D0%B4%D1%8D_%D0%B4%D1%8D%D0%B1%D0%B6%D2%AF%D2%AF%D0%BB%D1%85%D1%8D" title="Зэргэдэ дэбжүүлхэ – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Зэргэдэ дэбжүүлхэ" 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/Potenciaci%C3%B3" title="Potenciació – katalánština" lang="ca" hreflang="ca" data-title="Potenciació" data-language-autonym="Català" data-language-local-name="katalánština" 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/%D8%AA%D9%88%D8%A7%D9%86_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="توان (ماتماتیک) – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="توان (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BF%D0%B0%D1%88%D1%82%D0%B0%D1%80%D1%83" title="Капаштару – čuvaština" lang="cv" hreflang="cv" data-title="Капаштару" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Esbonydd" title="Esbonydd – velština" lang="cy" hreflang="cy" data-title="Esbonydd" data-language-autonym="Cymraeg" data-language-local-name="velština" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Potens_(matematik)" title="Potens (matematik) – dánština" lang="da" hreflang="da" data-title="Potens (matematik)" data-language-autonym="Dansk" data-language-local-name="dánština" 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/Potenz_(Mathematik)" title="Potenz (Mathematik) – němčina" lang="de" hreflang="de" data-title="Potenz (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="němčina" 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%94%CF%8D%CE%BD%CE%B1%CE%BC%CE%B7_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Δύναμη (μαθηματικά) – řečtina" lang="el" hreflang="el" data-title="Δύναμη (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" 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/Exponentiation" title="Exponentiation – angličtina" lang="en" hreflang="en" data-title="Exponentiation" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Potenco_(matematiko)" title="Potenco (matematiko) – esperanto" lang="eo" hreflang="eo" data-title="Potenco (matematiko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Potenciaci%C3%B3n" title="Potenciación – španělština" lang="es" hreflang="es" data-title="Potenciación" data-language-autonym="Español" data-language-local-name="španělština" 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/Astendamine" title="Astendamine – estonština" lang="et" hreflang="et" data-title="Astendamine" data-language-autonym="Eesti" data-language-local-name="estonština" 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/Berreketa" title="Berreketa – baskičtina" lang="eu" hreflang="eu" data-title="Berreketa" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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%AA%D9%88%D8%A7%D9%86_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="توان (ریاضی) – perština" lang="fa" hreflang="fa" data-title="توان (ریاضی)" data-language-autonym="فارسی" data-language-local-name="perština" 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/Potenssi" title="Potenssi – finština" lang="fi" hreflang="fi" data-title="Potenssi" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Potensur" title="Potensur – faerština" lang="fo" hreflang="fo" data-title="Potensur" data-language-autonym="Føroyskt" data-language-local-name="faerština" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Exponentiation" title="Exponentiation – francouzština" lang="fr" hreflang="fr" data-title="Exponentiation" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Potens" title="Potens – fríština (severní)" lang="frr" hreflang="frr" data-title="Potens" data-language-autonym="Nordfriisk" data-language-local-name="fríština (severní)" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Easp%C3%B3nant" title="Easpónant – irština" lang="ga" hreflang="ga" data-title="Easpónant" data-language-autonym="Gaeilge" data-language-local-name="irština" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%86%AA" title="冪 – čínština (dialekty Gan)" lang="gan" hreflang="gan" data-title="冪" data-language-autonym="贛語" data-language-local-name="čínština (dialekty Gan)" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Eksponansyasyon" title="Eksponansyasyon – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Eksponansyasyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Potenciaci%C3%B3n" title="Potenciación – galicijština" lang="gl" hreflang="gl" data-title="Potenciación" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he badge-Q17437796 badge-featuredarticle mw-list-item" title="nejlepší článek"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%96%D7%A7%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="חזקה (מתמטיקה) – hebrejština" lang="he" hreflang="he" data-title="חזקה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebrejština" 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%98%E0%A4%BE%E0%A4%A4%E0%A4%BE%E0%A4%82%E0%A4%95" title="घातांक – hindština" lang="hi" hreflang="hi" data-title="घातांक" data-language-autonym="हिन्दी" data-language-local-name="hindština" 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/Potenciranje" title="Potenciranje – chorvatština" lang="hr" hreflang="hr" data-title="Potenciranje" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" 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/Hatv%C3%A1ny" title="Hatvány – maďarština" lang="hu" hreflang="hu" data-title="Hatvány" data-language-autonym="Magyar" data-language-local-name="maďarština" 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%D5%BD%D5%BF%D5%AB%D5%B3%D5%A1%D5%B6_(%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE)" title="Աստիճան (հանրահաշիվ) – arménština" lang="hy" hreflang="hy" data-title="Աստիճան (հանրահաշիվ)" data-language-autonym="Հայերեն" data-language-local-name="arménština" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Potentiation" title="Potentiation – interlingua" lang="ia" hreflang="ia" data-title="Potentiation" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Eksponensiasi" title="Eksponensiasi – indonéština" lang="id" hreflang="id" data-title="Eksponensiasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Potenco" title="Potenco – ido" lang="io" hreflang="io" data-title="Potenco" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Veldi_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Veldi (stærðfræði) – islandština" lang="is" hreflang="is" data-title="Veldi (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandština" 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/Potenza_(matematica)" title="Potenza (matematica) – italština" lang="it" hreflang="it" data-title="Potenza (matematica)" data-language-autonym="Italiano" data-language-local-name="italština" 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%E4%B9%97" title="冪乗 – japonština" lang="ja" hreflang="ja" data-title="冪乗" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Exponenshieshan" title="Exponenshieshan – jamajská kreolština" lang="jam" hreflang="jam" data-title="Exponenshieshan" data-language-autonym="Patois" data-language-local-name="jamajská kreolština" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D3%99%D1%80%D0%B5%D0%B6%D0%B5%D0%BB%D0%B5%D1%83" title="Дәрежелеу – kazaština" lang="kk" hreflang="kk" data-title="Дәрежелеу" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</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" title="거듭제곱 – korejština" lang="ko" hreflang="ko" data-title="거듭제곱" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Potentia_(mathematica)" title="Potentia (mathematica) – latina" lang="la" hreflang="la" data-title="Potentia (mathematica)" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Esponenti" title="Esponenti – lingua franca nova" lang="lfn" hreflang="lfn" data-title="Esponenti" data-language-autonym="Lingua Franca Nova" data-language-local-name="lingua franca nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Machsverh%C3%B6ffing" title="Machsverhöffing – limburština" lang="li" hreflang="li" data-title="Machsverhöffing" data-language-autonym="Limburgs" data-language-local-name="limburština" 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/K%C4%97limas_laipsniu" title="Kėlimas laipsniu – litevština" lang="lt" hreflang="lt" data-title="Kėlimas laipsniu" data-language-autonym="Lietuvių" data-language-local-name="litevština" 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/K%C4%81pin%C4%81%C5%A1ana" title="Kāpināšana – lotyština" lang="lv" hreflang="lv" data-title="Kāpināšana" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Toraka_(matematika)" title="Toraka (matematika) – malgaština" lang="mg" hreflang="mg" data-title="Toraka (matematika)" data-language-autonym="Malagasy" data-language-local-name="malgaština" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Степенување – makedonština" lang="mk" hreflang="mk" data-title="Степенување" data-language-autonym="Македонски" data-language-local-name="makedonština" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pengeksponenan" title="Pengeksponenan – malajština" lang="ms" hreflang="ms" data-title="Pengeksponenan" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%98%E0%A4%BE%E0%A4%A4%E0%A4%BE%E0%A4%99%E0%A5%8D%E0%A4%95" title="घाताङ्क – nepálština" lang="ne" hreflang="ne" data-title="घाताङ्क" data-language-autonym="नेपाली" data-language-local-name="nepálština" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Machtsverheffen" title="Machtsverheffen – nizozemština" lang="nl" hreflang="nl" data-title="Machtsverheffen" data-language-autonym="Nederlands" data-language-local-name="nizozemština" 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/Potens_i_matematikk" title="Potens i matematikk – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Potens i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (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/Potens_(matematikk)" title="Potens (matematikk) – norština (bokmål)" lang="nb" hreflang="nb" data-title="Potens (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norština (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/Aangessoo(ekispoonentii)" title="Aangessoo(ekispoonentii) – oromština" lang="om" hreflang="om" data-title="Aangessoo(ekispoonentii)" data-language-autonym="Oromoo" data-language-local-name="oromština" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%98%E0%A8%BE%E0%A8%A4_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਘਾਤ ਅੰਕ – paňdžábština" lang="pa" hreflang="pa" data-title="ਘਾਤ ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="paňdžábština" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pot%C4%99gowanie" title="Potęgowanie – polština" lang="pl" hreflang="pl" data-title="Potęgowanie" data-language-autonym="Polski" data-language-local-name="polština" 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/Exponencia%C3%A7%C3%A3o" title="Exponenciação – portugalština" lang="pt" hreflang="pt" data-title="Exponenciação" data-language-autonym="Português" data-language-local-name="portugalština" 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/Yupa_huqariy" title="Yupa huqariy – kečuánština" lang="qu" hreflang="qu" data-title="Yupa huqariy" data-language-autonym="Runa Simi" data-language-local-name="kečuánština" 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/Putere_(matematic%C4%83)" title="Putere (matematică) – rumunština" lang="ro" hreflang="ro" data-title="Putere (matematică)" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BE%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B2_%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8C" title="Возведение в степень – ruština" lang="ru" hreflang="ru" data-title="Возведение в степень" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%91%D2%AF%D1%82%D2%AF%D0%BD_%D0%BA%D3%A9%D1%80%D0%B4%D3%A9%D1%80%D3%A9%D3%A9%D1%87%D1%87%D2%AF%D0%BB%D1%8D%D1%8D%D1%85_%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8C" title="Бүтүн көрдөрөөччүлээх степень – jakutština" lang="sah" hreflang="sah" data-title="Бүтүн көрдөрөөччүлээх степень" data-language-autonym="Саха тыла" data-language-local-name="jakutština" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Putenza_(matim%C3%A0tica)" title="Putenza (matimàtica) – sicilština" lang="scn" hreflang="scn" data-title="Putenza (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="sicilština" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Stepenovanje" title="Stepenovanje – srbochorvatština" lang="sh" hreflang="sh" data-title="Stepenovanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Exponentiation" title="Exponentiation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Exponentiation" 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/Umoc%C5%88ovanie" title="Umocňovanie – slovenština" lang="sk" hreflang="sk" data-title="Umocňovanie" data-language-autonym="Slovenčina" data-language-local-name="slovenština" 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/Potenciranje" title="Potenciranje – slovinština" lang="sl" hreflang="sl" data-title="Potenciranje" data-language-autonym="Slovenščina" data-language-local-name="slovinština" 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/Kutambanura_(nhamba)" title="Kutambanura (nhamba) – šonština" lang="sn" hreflang="sn" data-title="Kutambanura (nhamba)" data-language-autonym="ChiShona" data-language-local-name="šonština" 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%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D0%BE%D0%B2%D0%B0%D1%9A%D0%B5" title="Степеновање – srbština" lang="sr" hreflang="sr" data-title="Степеновање" data-language-autonym="Српски / srpski" data-language-local-name="srbština" 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/Potens" title="Potens – švédština" lang="sv" hreflang="sv" data-title="Potens" data-language-autonym="Svenska" data-language-local-name="švédština" 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/%E0%AE%85%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="அடுக்கேற்றம் – tamilština" lang="ta" hreflang="ta" data-title="அடுக்கேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th badge-Q17437798 badge-goodarticle mw-list-item" title="dobrý článek"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A2%E0%B8%81%E0%B8%81%E0%B8%B3%E0%B8%A5%E0%B8%B1%E0%B8%87" title="การยกกำลัง – thajština" lang="th" hreflang="th" data-title="การยกกำลัง" data-language-autonym="ไทย" data-language-local-name="thajština" 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/Pagpapalakas_(matematika)" title="Pagpapalakas (matematika) – tagalog" lang="tl" hreflang="tl" data-title="Pagpapalakas (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9Cs" title="Üs – turečtina" lang="tr" hreflang="tr" data-title="Üs" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D8%AF%DB%95%D8%B1%D9%89%D8%AC%DB%95_(%D9%85%D8%A7%D8%AA%DB%90%D9%85%D8%A7%D8%AA%D9%89%D9%83%D8%A7)" title="دەرىجە (ماتېماتىكا) – ujgurština" lang="ug" hreflang="ug" data-title="دەرىجە (ماتېماتىكا)" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="ujgurština" 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%9F%D1%96%D0%B4%D0%BD%D0%B5%D1%81%D0%B5%D0%BD%D0%BD%D1%8F_%D0%B4%D0%BE_%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D1%8F" title="Піднесення до степеня – ukrajinština" lang="uk" hreflang="uk" data-title="Піднесення до степеня" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Darajaga_ko%CA%BBtarish" title="Darajaga koʻtarish – uzbečtina" lang="uz" hreflang="uz" data-title="Darajaga koʻtarish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" 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/L%C5%A9y_th%E1%BB%ABa" title="Lũy thừa – vietnamština" lang="vi" hreflang="vi" data-title="Lũy thừa" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" 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/Eksponentasyon" title="Eksponentasyon – warajština" lang="war" hreflang="war" data-title="Eksponentasyon" data-language-autonym="Winaray" data-language-local-name="warajština" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a 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href="https://commons.wikimedia.org/wiki/Category:Exponentiation" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikifunctions mw-list-item"><a href="https://www.wikifunctions.org/wiki/Z12665" hreflang="en"><span>Wikifunkce</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q33456" title="Odkaz na propojenou položku datového úložiště [g]" accesskey="g"><span>Položka Wikidat</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Nástroje ke stránce"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Vzhled"> <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">Vzhled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skrýt</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">Z Wikipedie, otevřené encyklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(přesměrováno z <a href="/w/index.php?title=Mocnina&redirect=no" class="mw-redirect" title="Mocnina">Mocnina</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"><p><b>Umocňování</b> je <a href="/wiki/Matematika" title="Matematika">matematická</a> <a href="/wiki/Operace_(matematika)" title="Operace (matematika)">operace</a>, která vychází z opakovaného <a href="/wiki/N%C3%A1soben%C3%AD" title="Násobení">násobení</a>. Umocňování je k násobení v podobném vztahu, v jakém je samo <a href="/wiki/N%C3%A1soben%C3%AD" title="Násobení">násobení</a> ke <a href="/wiki/S%C4%8D%C3%ADt%C3%A1n%C3%AD" title="Sčítání">sčítání</a>. Umocňování slouží ke zkrácenému zápisu vícenásobného násobení: </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 \underbrace {z\cdot z\cdot z\cdots z} _{n\operatorname {-kr{\acute {a}}t} }=z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>z</mi> <mo>⋅</mo> <mi>z</mi> <mo>⋅</mo> <mi>z</mi> <mo>⋯</mo> <mi>z</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mtext>-</mtext> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">a</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">t</mi> </mrow> </mrow> </munder> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {z\cdot z\cdot z\cdots z} _{n\operatorname {-kr{\acute {a}}t} }=z^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9f96e19be6c72e3b5bdafd195c8f26cd3965c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:16.615ex; height:6.176ex;" alt="{\displaystyle \underbrace {z\cdot z\cdot z\cdots z} _{n\operatorname {-kr{\acute {a}}t} }=z^{n}}" /></span></dd></dl> <p>V tomto vzorci se <var>z</var> označuje jako <b>základ mocniny</b> (mocněnec) a <var>n</var> se nazývá <b>exponent</b> (mocnitel). Výsledek je „<var>n</var>-tá <b>mocnina</b> čísla <var>z</var>“, „<var>z</var> na <var>n</var>-tou“. Například <span style="white-space:nowrap">3 · 3 · 3 · 3 = 81</span> je „tři na čtvrtou“, což zapisujeme 3<sup>4</sup>. Exponent může být obecně reálné, nebo dokonce komplexní číslo (viz <a href="#Definice">#Definice</a>). </p><p>Speciálním případem prázdného součinu je <var>z</var><sup>0</sup> = 1 (pro <var>z</var> ≠ 0, jinak viz <a href="#Nula_na_nultou">#Nula na nultou</a>). Pro nulový základ a kladný exponent (<var>n</var> > 0) pak platí 0<sup><var>n</var></sup> = 0. </p><p>Když z technických důvodů nelze psát exponent na horní pozici, používá se často zápis ve tvaru <code>z^n</code>, někdy také <code>z**n</code>. </p><p>Pomocí umocňování je definováno několik základních <a href="/wiki/Funkce_(matematika)" title="Funkce (matematika)">funkcí</a> a <a href="/wiki/Posloupnost" title="Posloupnost">posloupností</a>: <a href="/wiki/Mocninn%C3%A1_funkce" title="Mocninná funkce">Mocninná funkce</a> <span style="white-space:nowrap"><var>f</var>(<var>x</var>) = <var>a</var> · <var>x</var><sup><var>n</var></sup></span>, <a href="/wiki/Exponenci%C3%A1ln%C3%AD_funkce" title="Exponenciální funkce">exponenciální funkce</a> <span style="white-space:nowrap"><var>f</var>(<var>x</var>) = <var>z</var><sup><var>x</var></sup></span>, <a href="/wiki/Geometrick%C3%A1_posloupnost" title="Geometrická posloupnost">geometrická posloupnost</a> <span style="white-space:nowrap"><var>a</var><sub><var>n</var></sub> = <var>z</var><sup><var>n</var></sup></span> a funkce <span style="white-space:nowrap"><var>f</var>(<var>x</var>) = <var>x</var><sup><var>x</var></sup></span>. </p><p><a href="/wiki/Inverzn%C3%AD_operace" class="mw-redirect" title="Inverzní operace">Inverzní operace</a> k umocňování je <a href="/wiki/Odmoc%C5%88ov%C3%A1n%C3%AD" class="mw-redirect" title="Odmocňování">odmocňování</a>. Opakované umocňování je <a href="/wiki/Tetrace" title="Tetrace">tetrace</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definice">Definice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=1" title="Editace sekce: Definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=1" title="Editovat zdrojový kód sekce Definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mocnina s přirozeným exponentem (<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\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" /></span>) se tedy definuje jako opakované násobení, které lze zapsat rekurentně takto: </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 z^{1}=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{1}=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3131305a33fcfdce504367ed44b1bb191c19291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.331ex; height:2.676ex;" alt="{\displaystyle z^{1}=z}" /></span></dd> <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 z^{n+1}=z^{n}\cdot z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n+1}=z^{n}\cdot z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9c1e138c14ae5e5e99c188586906778a7de390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.584ex; height:2.676ex;" alt="{\displaystyle z^{n+1}=z^{n}\cdot z}" /></span></dd></dl> <p>Rekurentní vzorec lze obrátit a tak při nenulovém základu (<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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8b7eb2d2a30057811a7835502717d3d6ece962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}" /></span>) tuto definici použít i pro ostatní celé exponenty (<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\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1cf6a513f2062531d95dbb198944936f312982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.786ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {Z} }" /></span>): </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 z^{n}={z^{n+1} \over z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>z</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}={z^{n+1} \over z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5922849fbca698e070379fce2a5ba8b00f32c2b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.653ex; height:5.676ex;" alt="{\displaystyle z^{n}={z^{n+1} \over z}}" /></span></dd></dl> <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 z^{0}={z \over z}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>z</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{0}={z \over z}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ade7af97fccb97d6927e86eadfddb81fd521c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.428ex; height:4.676ex;" alt="{\displaystyle z^{0}={z \over z}=1}" /></span></dd></dl> <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 z^{-n}={1 \over z^{n}}=\left({1 \over z}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{-n}={1 \over z^{n}}=\left({1 \over z}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219e0e10e8ad0a6e25e41342488cac95934c81e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.567ex; height:6.176ex;" alt="{\displaystyle z^{-n}={1 \over z^{n}}=\left({1 \over z}\right)^{n}}" /></span></dd></dl> <p>Definici lze dále zobecnit pro <a href="/wiki/Racion%C3%A1ln%C3%AD_%C4%8D%C3%ADslo" title="Racionální číslo">racionální</a> exponent s využitím <a href="/wiki/Odmoc%C5%88ov%C3%A1n%C3%AD" class="mw-redirect" title="Odmocňování">odmocňování</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 z^{n \over m}={\sqrt[{m}]{z^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n \over m}={\sqrt[{m}]{z^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3cb5a6f24d4c730ee30852bfc0c056f5f7744f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.673ex; height:3.343ex;" alt="{\displaystyle z^{n \over m}={\sqrt[{m}]{z^{n}}}}" /></span></dd></dl> <p>Zobecnění na celý obor <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálných čísel</a> (tzn. rozšíření definice o mocniny s <a href="/wiki/Iracion%C3%A1ln%C3%AD_%C4%8D%C3%ADslo" title="Iracionální číslo">iracionálními</a> exponenty) se pak dosahuje dodefinováním pomocí <a href="/wiki/Limita" title="Limita">limity</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 z^{n}=\lim _{x\to n \atop x\in \mathbb {Q} }z^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi>n</mi> </mrow> <mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </mfrac> </mrow> </munder> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=\lim _{x\to n \atop x\in \mathbb {Q} }z^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f110ade201fbf617d2e32ac9927b6ef11cfdcf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.513ex; height:5.176ex;" alt="{\displaystyle z^{n}=\lim _{x\to n \atop x\in \mathbb {Q} }z^{x}}" /></span></dd></dl> <p>Pro mocniny s <a href="/wiki/Komplexn%C3%AD_%C4%8D%C3%ADslo" title="Komplexní číslo">komplexním</a> základem <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=a+bi=r\cdot (\cos \varphi +i\sin \varphi )=r\cdot e^{i\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+bi=r\cdot (\cos \varphi +i\sin \varphi )=r\cdot e^{i\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41fb04f162d910b45326b4a4d3a1b84220c49519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.288ex; height:3.176ex;" alt="{\displaystyle z=a+bi=r\cdot (\cos \varphi +i\sin \varphi )=r\cdot e^{i\varphi }}" /></span>, kde <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,b,\varphi \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>φ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,\varphi \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47de88bfc22b8ef5617160f0ce59a44578858455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.334ex; height:2.676ex;" alt="{\displaystyle a,b,\varphi \in \mathbb {R} }" /></span> 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 r\in \mathbb {R} _{0}^{+},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {R} _{0}^{+},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f242a70f27cc1d822e42256381bb5bbd442bcaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.725ex; height:3.176ex;" alt="{\displaystyle r\in \mathbb {R} _{0}^{+},}" /></span> <span style="white-space:nowrap">pak platí (viz <a href="/wiki/Moivrova_v%C4%9Bta" class="mw-redirect" title="Moivrova věta">Moivrovu větu</a>)</span> </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 z^{n}\equiv (a+bi)^{n}=(r\cdot e^{i\varphi })^{n}=r^{n}\cdot e^{i\;n\varphi }=r^{n}\cdot [\cos(n\varphi )+i\sin(n\varphi )].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>≡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thickmathspace"></mspace> <mi>n</mi> <mi>φ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}\equiv (a+bi)^{n}=(r\cdot e^{i\varphi })^{n}=r^{n}\cdot e^{i\;n\varphi }=r^{n}\cdot [\cos(n\varphi )+i\sin(n\varphi )].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4d72c13978b0767e34d1ef290c14546c9e7350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.182ex; height:3.176ex;" alt="{\displaystyle z^{n}\equiv (a+bi)^{n}=(r\cdot e^{i\varphi })^{n}=r^{n}\cdot e^{i\;n\varphi }=r^{n}\cdot [\cos(n\varphi )+i\sin(n\varphi )].}" /></span></dd></dl> <p><a href="/wiki/Argument_(komplexn%C3%AD_anal%C3%BDza)" class="mw-redirect" title="Argument (komplexní analýza)">Argument</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 \varphi =\operatorname {Arg} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>Arg</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\operatorname {Arg} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeb963dea27d22be2c5dc9fc46f9c938afea34c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.911ex; height:2.676ex;" alt="{\displaystyle \varphi =\operatorname {Arg} z}" /></span> má nutně skok, jehož polohu však lze zvolit. Volí se zpravidla <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> z intervalu <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 \langle 0;2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>;</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0;2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab0278b12c3c7145e6bfb4db53e77c993fe26f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle \langle 0;2\pi )}" /></span> nebo <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 (-\pi ;\pi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>;</mo> <mi>π</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ;\pi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b72fcd9b209ab8b64ea846cf870f148a5c5091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.315ex; height:2.843ex;" alt="{\displaystyle (-\pi ;\pi \rangle }" /></span>. Komplexní mocnina s neceločíselným exponentem je tedy obecně <a href="/w/index.php?title=Mnohozna%C4%8Dn%C3%A1_funkce&action=edit&redlink=1" class="new" title="Mnohoznačná funkce (stránka neexistuje)">mnohoznačná funkce</a> a není na celé <a href="/wiki/Komplexn%C3%AD_rovina" title="Komplexní rovina">komplexní rovině</a> <a href="/wiki/Holomorfn%C3%AD_funkce" title="Holomorfní funkce">holomorfní</a>. </p><p>Pokud je navíc komplexním číslem i exponent <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>, pak je mocnina dána jako </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 z^{n}=e^{n\ln z}=e^{n(i\varphi +\ln r)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mi>φ</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=e^{n\ln z}=e^{n(i\varphi +\ln r)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a121dd6b44d4e2dfacde62fe9fb1f8e8c0fa3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.372ex; height:2.843ex;" alt="{\displaystyle z^{n}=e^{n\ln z}=e^{n(i\varphi +\ln r)}.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Alternativní_definice"><span id="Alternativn.C3.AD_definice"></span>Alternativní definice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=2" title="Editace sekce: Alternativní definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=2" title="Editovat zdrojový kód sekce Alternativní definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Užitečná definice z oblasti <a href="/wiki/Teorie_mno%C5%BEin" title="Teorie množin">teorie množin</a> říká, že pro množiny <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,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" /></span> je <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^{B}=\{f|f:B\rightarrow A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{B}=\{f|f:B\rightarrow A\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a51733d8f9e57aebd585e5720fa92d311adb3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.908ex; height:3.176ex;" alt="{\displaystyle A^{B}=\{f|f:B\rightarrow A\}}" /></span> čili množina všech <a href="/wiki/Zobrazen%C3%AD_(matematika)" title="Zobrazení (matematika)">zobrazení</a> množiny <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> do množiny <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>, tedy takových zobrazení, která každému prvku z <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /></span> přiřazují právě jeden prvek z <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span>. Jsou-li obě množiny konečné, pak počet takových zobrazení je <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^{B}\right|=|A|^{|B|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msup> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|A^{B}\right|=|A|^{|B|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0d31a5d12871cd591cd673492a33015a8ac741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.046ex; height:3.843ex;" alt="{\displaystyle \left|A^{B}\right|=|A|^{|B|}}" /></span>, přičemž klademe 0<sup>0</sup> = 1 (viz <a href="#Nula_na_nultou">#Nula na nultou</a>). Příklad: </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 \{0,1\}^{\{a,b\}}={\Big \{}\{a\mapsto 0;b\mapsto 0\},\{a\mapsto 0;b\mapsto 1\},\{a\mapsto 1;b\mapsto 0\},\{a\mapsto 1;b\mapsto 1\}{\Big \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mn>0</mn> <mo>;</mo> <mi>b</mi> <mo stretchy="false">↦</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mn>0</mn> <mo>;</mo> <mi>b</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mo>;</mo> <mi>b</mi> <mo stretchy="false">↦</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mo>;</mo> <mi>b</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1\}^{\{a,b\}}={\Big \{}\{a\mapsto 0;b\mapsto 0\},\{a\mapsto 0;b\mapsto 1\},\{a\mapsto 1;b\mapsto 0\},\{a\mapsto 1;b\mapsto 1\}{\Big \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a11f9b8141518fb5d698bf1600f83128f4baa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:79.45ex; height:4.843ex;" alt="{\displaystyle \{0,1\}^{\{a,b\}}={\Big \{}\{a\mapsto 0;b\mapsto 0\},\{a\mapsto 0;b\mapsto 1\},\{a\mapsto 1;b\mapsto 0\},\{a\mapsto 1;b\mapsto 1\}{\Big \}}}" /></span></dd></dl> <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 \left|\{0,1\}^{\{a,b\}}\right|=|\{0,1\}|^{|\{a,b\}|}=2^{2}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\{0,1\}^{\{a,b\}}\right|=|\{0,1\}|^{|\{a,b\}|}=2^{2}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8d5806f6676424956ef0a88106213962bf4912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.361ex; height:4.009ex;" alt="{\displaystyle \left|\{0,1\}^{\{a,b\}}\right|=|\{0,1\}|^{|\{a,b\}|}=2^{2}=4}" /></span></dd></dl> <p>Mocninu <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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1a8cdd7ee39054e510deeb38ee551cc7616ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle z^{n}}" /></span> s nezáporným celým základem i exponentem (<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,n\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z,n\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bc49583ddd506182d1b643fd3ddca908bd2c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.09ex; height:2.509ex;" alt="{\displaystyle z,n\in \mathbb {N} _{0}}" /></span>) lze také vyjádřit jako počet všech uspořádaných <span style="white-space:nowrap"><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>-tic,</span> jejichž složky jsou ze <span style="white-space:nowrap"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span>-prvkové</span> množiny. Toto vyjádření je velmi podobné předchozí definici, protože zobrazení <span style="white-space:nowrap"><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>-prvkové</span> množiny lze zapsat jako uspořádanou <span style="white-space:nowrap"><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>-tici.</span> Příklad: </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 2^{3}=\left|\{0,1\}^{3}\right|=\left|{\Big \{}(0,0,0),(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,1,0),(1,0,1),(1,1,1){\Big \}}\right|=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}=\left|\{0,1\}^{3}\right|=\left|{\Big \{}(0,0,0),(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,1,0),(1,0,1),(1,1,1){\Big \}}\right|=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a2b8480453866614d3b6195ec0d6d5523a9e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:91.254ex; height:4.843ex;" alt="{\displaystyle 2^{3}=\left|\{0,1\}^{3}\right|=\left|{\Big \{}(0,0,0),(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,1,0),(1,0,1),(1,1,1){\Big \}}\right|=8}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=3" title="Editace sekce: Vlastnosti" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=3" title="Editovat zdrojový kód sekce Vlastnosti"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pro <a href="/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo">reálná</a> nebo <a href="/wiki/Komplexn%C3%AD_%C4%8D%C3%ADsla" class="mw-redirect" title="Komplexní čísla">komplexní čísla</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 a,b,x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2248914b9e42600ee8c4df5e8b347baa934642ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.814ex; height:2.509ex;" alt="{\displaystyle a,b,x,y}" /></span> platí následující vztahy (jsou-li výrazy na obou stranách definované): </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 \left(ab\right)^{x}=a^{x}\cdot b^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(ab\right)^{x}=a^{x}\cdot b^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df58f1dfd7c4a49f4a340c95fce42683fd730af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:14.559ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle \left(ab\right)^{x}=a^{x}\cdot b^{x}}" /></span> za podmínky, že <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> je <a href="/wiki/Cel%C3%A9_%C4%8D%C3%ADslo" title="Celé číslo">celé číslo</a> nebo <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 \operatorname {Arg} a+\operatorname {Arg} b\in (-\pi ;\pi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Arg</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>Arg</mi> <mo>⁡</mo> <mi>b</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>;</mo> <mi>π</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} a+\operatorname {Arg} b\in (-\pi ;\pi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70dc716fbb8f979fad00ccd41e5ffbeeab3b85e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.632ex; height:2.843ex;" alt="{\displaystyle \operatorname {Arg} a+\operatorname {Arg} b\in (-\pi ;\pi \rangle }" /></span>, tedy že se neprojeví skok <a href="/wiki/Argument_(komplexn%C3%AD_anal%C3%BDza)" class="mw-redirect" title="Argument (komplexní analýza)">argumentu</a></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({\frac {a}{b}}\right)^{x}={\frac {a^{x}}{b^{x}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {a}{b}}\right)^{x}={\frac {a^{x}}{b^{x}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac46d0b5e7ed8a1ad2ddd45e0f0f4dac746e5cef" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -2.005ex; width:12.351ex; height:5.343ex;" aria-hidden="true" alt="{\displaystyle \left({\frac {a}{b}}\right)^{x}={\frac {a^{x}}{b^{x}}}}" /></span> za podmínky, že <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> je celé číslo nebo <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 \operatorname {Arg} a-\operatorname {Arg} b\in (-\pi ;\pi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Arg</mi> <mo>⁡</mo> <mi>a</mi> <mo>−</mo> <mi>Arg</mi> <mo>⁡</mo> <mi>b</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>;</mo> <mi>π</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} a-\operatorname {Arg} b\in (-\pi ;\pi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1270683e6e9d6ecfc2af48c6ffc5da59022d9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.632ex; height:2.843ex;" alt="{\displaystyle \operatorname {Arg} a-\operatorname {Arg} b\in (-\pi ;\pi \rangle }" /></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 a^{x}\cdot a^{y}=a^{x+y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{x}\cdot a^{y}=a^{x+y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe074be132037bc3c3e8abf1c61fe2d53b929fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.957ex; height:2.509ex;" alt="{\displaystyle a^{x}\cdot a^{y}=a^{x+y}}" /></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 a^{-x}={\frac {1}{a^{x}}},\quad a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-x}={\frac {1}{a^{x}}},\quad a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e61e915c56a03a2f0ce4bafc46b4ad3a5305d1af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.865ex; height:5.176ex;" alt="{\displaystyle a^{-x}={\frac {1}{a^{x}}},\quad a\neq 0}" /></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 {a^{x}}{a^{y}}}=a^{x-y},\quad a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em"></mspace> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a^{x}}{a^{y}}}=a^{x-y},\quad a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f4a2c8e9dbeaeb9b7f4900e429203781f78d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.682ex; height:5.176ex;" alt="{\displaystyle {\frac {a^{x}}{a^{y}}}=a^{x-y},\quad a\neq 0}" /></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(a^{x}\right)^{y}=a^{x\cdot y}}"> <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>x</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a^{x}\right)^{y}=a^{x\cdot y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/005c070ce95e5a9ac09a0acd99ea11f52b20161d" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:12.036ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle \left(a^{x}\right)^{y}=a^{x\cdot y}}" /></span> za podmínky, že <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> je celé číslo nebo <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 \operatorname {Im} (x\ln a)\in (-\pi ;\pi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>;</mo> <mi>π</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (x\ln a)\in (-\pi ;\pi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06a2703cd822dea5ec4b21a36fa997543444080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.014ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (x\ln a)\in (-\pi ;\pi \rangle }" /></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 a^{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/448ca9a3f4ef03c4dfcf69258912d2c90b097842" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:6.545ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle a^{0}=1}" /></span> pro <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" /></span> (pro 0<sup>0</sup> viz <a href="#Nula_na_nultou">níže</a>)</li></ul> <p>Umocňování není obecně <a href="/wiki/Komutativita" title="Komutativita">komutativní</a> (2<sup>3</sup> ≠ 3<sup>2</sup>) ani <a href="/wiki/Asociativita" title="Asociativita">asociativní</a>: (2<sup>2</sup>)<sup>3</sup> ≠ 2<sup>(2<sup>3</sup>)</sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Mocniny_nuly">Mocniny nuly</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=4" title="Editace sekce: Mocniny nuly" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=4" title="Editovat zdrojový kód sekce Mocniny nuly"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nula umocněná na kladné číslo je nula, tedy pro <i>x</i> > 0 je 0<sup><i>x</i></sup> = 0. </p><p>Naproti tomu nula umocněná na záporné číslo není definována, protože takový výraz vede na <a href="/wiki/D%C4%9Blen%C3%AD_nulou" title="Dělení nulou">dělení nulou</a>, které není na množině reálných ani komplexních čísel definováno: </p> <dl><dd>Pro <span class="texhtml"><var>x</var></span> > 0 je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{-x}={1 \over 0^{x}}={1 \over 0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>0</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{-x}={1 \over 0^{x}}={1 \over 0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4292683d531d3470ff9d8dd8f286a56b11e9ef2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -2.005ex; width:15.627ex; height:5.343ex;" aria-hidden="true" alt="{\displaystyle 0^{-x}={1 \over 0^{x}}={1 \over 0}.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Nula_na_nultou">Nula na nultou</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=5" title="Editace sekce: Nula na nultou" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=5" title="Editovat zdrojový kód sekce Nula na nultou"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zcela obecně není výraz 0<sup>0</sup> definován. <a href="/wiki/Limita" title="Limita">Limita</a> mocniny, jejíž základ i exponent konvergují k nule, je totiž tzv. <a href="/w/index.php?title=Neur%C4%8Dit%C3%BD_v%C3%BDraz&action=edit&redlink=1" class="new" title="Neurčitý výraz (stránka neexistuje)">neurčitý výraz</a> a pro její vyčíslení je potřeba znát vztah mezi základem a exponentem. Na výraz 0<sup>0</sup> se tedy lze dívat dvěma základními způsoby. První pohled na něj hledí jako na limitu funkce <i>x</i><sup>0</sup>, která je všude kromě nuly rovna jedné, takže je možno ji v nule dodefinovat stejně a klade se 0<sup>0</sup> = 1. Naopak druhý pohled vychází z funkce 0<sup><i>x</i></sup>, která je pro všechna kladná <i>x</i> nulová, takže se i v nule dodefinuje 0<sup>0</sup> = 0. </p><p>V běžných situacích se používá hlavně první definice (0<sup>0</sup> = 1),<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> která je vyžadována pro jednoduchý zápis mnoha vzorců: </p> <ul><li>Aby při zápisu <a href="/wiki/Polynom" title="Polynom">polynomu</a> ve tvaru <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(x)=\sum _{k=0}^{n}{a_{k}x^{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=\sum _{k=0}^{n}{a_{k}x^{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23f3b912a69662056965d72962ed52daf0acb264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:15.975ex; height:7.009ex;" alt="{\displaystyle p(x)=\sum _{k=0}^{n}{a_{k}x^{k}}}" /></span> platilo <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(0)=a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(0)=a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da401f5b044dbfa0b885e45515d6fd426771c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.613ex; height:2.843ex;" alt="{\displaystyle p(0)=a_{0}}" /></span>, musí být 0<sup>0</sup> = 1. Podobný zápis se používá také pro <a href="/wiki/Mocninn%C3%A1_%C5%99ada" title="Mocninná řada">mocninnou řadu</a>.</li> <li>Obecná platnost <a href="/wiki/Binomick%C3%A1_v%C4%9Bta" title="Binomická věta">binomické věty</a> vyžaduje 0<sup>0</sup> = 1.<sup id="cite_ref-knuth_2-0" class="reference"><a href="#cite_note-knuth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>Existuje právě jedno zobrazení prázdné množiny do prázdné množiny, a to prázdné zobrazení (viz <a href="#Alternativní_definice">#Alternativní definice</a>).</li> <li>Pravidlo pro <a href="/wiki/Derivov%C3%A1n%C3%AD" class="mw-redirect" title="Derivování">derivování</a> <a href="/wiki/Mocninn%C3%A1_funkce" title="Mocninná funkce">mocninné funkce</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 {\tfrac {\operatorname {d} x^{n}}{\operatorname {d} x}}=nx^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>n</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\operatorname {d} x^{n}}{\operatorname {d} x}}=nx^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c55fbb0629c1d6cb5e06bbb081123e9eac7b051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.184ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\operatorname {d} x^{n}}{\operatorname {d} x}}=nx^{n-1}}" /></span> platí pro <span class="texhtml"><var>n</var></span> = 1 v bodě <span class="texhtml"><var>x</var></span> = 0 jen tehdy, když 0<sup>0</sup> = 1.</li></ul> <p>Jindy je 0<sup>0</sup> ponecháno nedefinované,<sup id="cite_ref-knuth_2-1" class="reference"><a href="#cite_note-knuth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> zcela výjimečně je možno se setkat i s použitím druhé definice (0<sup>0</sup> = 0).<sup class="noexcerpt" style="white-space: pre;">[<a href="/wiki/Wikipedie:Ov%C4%9B%C5%99itelnost" title="Wikipedie:Ověřitelnost"><span class="doplnte-zdroj" title="Je potřeba doložit zdroj předchozího tvrzení, označeno 24. 4. 2015 13:32:50 CEST">zdroj⁠?!</span></a>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Zvláštní_mocniny"><span id="Zvl.C3.A1.C5.A1tn.C3.AD_mocniny"></span>Zvláštní mocniny</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=6" title="Editace sekce: Zvláštní mocniny" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=6" title="Editovat zdrojový kód sekce Zvláštní mocniny"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>V každodenním životě často používáme mocniny o základu deset (to jsou 1, 10, 100, 1000, …). Tyto mocniny tvoří základ naší desítkové <a href="/wiki/%C4%8C%C3%ADseln%C3%A1_soustava" title="Číselná soustava">číselné soustavy</a>, také v <a href="/wiki/Soustava_SI" title="Soustava SI">soustavě SI</a> jsou <a href="/wiki/P%C5%99edpona_soustavy_SI" title="Předpona soustavy SI">předpony</a> násobků jednotek označením mocnin deseti – 1 kg = 10³ g apod. </p><p>Velmi časté je rovněž využití druhé mocniny (a<sup>2</sup>), tj. vynásobení čísla <i>a</i> sama sebou. <a href="/wiki/Druh%C3%A1_mocnina" title="Druhá mocnina">Druhá mocnina</a> je v běžné řeči někdy označována jako <b>čtverec</b>, protože obsah <a href="/wiki/%C4%8Ctverec" title="Čtverec">čtverce</a> je roven druhé mocnině délky jeho hrany (S = a<sup>2</sup>). </p><p><a href="/wiki/Po%C4%8D%C3%ADta%C4%8D" title="Počítač">Počítače</a> při zpracování dat používají <a href="/wiki/Dvojkov%C3%A1_soustava" title="Dvojková soustava">dvojkovou soustavu</a>, založenou na mocninách čísla 2. Z toho důvodu se někdy v <a href="/wiki/Informatika" title="Informatika">informatice</a> používají násobky jednotek jako mocniny o základu 2 – 1 KiB = 2<sup>10</sup> B = 1024 B. (Viz též <a href="/wiki/Bin%C3%A1rn%C3%AD_p%C5%99edpony" class="mw-redirect" title="Binární předpony">binární předpony</a>.) </p><p>V matematice jsou zvlášť důležité mocniny o základu <i>e</i> ≅ 2,71828, takzvaného <a href="/wiki/Eulerovo_%C4%8D%C3%ADslo" title="Eulerovo číslo">Eulerova čísla</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Reference">Reference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=7" title="Editace sekce: Reference" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=7" title="Editovat zdrojový kód sekce Reference"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Všechny následující výpočetní prostředky poskytují výsledek 0<sup>0</sup> = 1: <a rel="nofollow" class="external text" href="https://www.google.cz/search?q=0^0">Vyhledávač Google</a>, Kalkulačka ve Windows 7, funkce <a rel="nofollow" class="external text" href="http://en.cppreference.com/w/cpp/numeric/math/pow">pow</a> jazyka C++, metoda <a rel="nofollow" class="external text" href="https://msdn.microsoft.com/en-US/library/system.math.pow%28v=vs.100%29.aspx">System.Math.Pow</a> z MS .NET Framework.</span> </li> <li id="cite_note-knuth-2"><span class="mw-cite-backlink">↑ <a href="#cite_ref-knuth_2-0"><sup style="font-style: italic; font-weight: bold; vertical-align: top">a</sup></a> <a href="#cite_ref-knuth_2-1"><sup style="font-style: italic; font-weight: bold; vertical-align: top">b</sup></a></span> <span class="reference-text"><cite style="font-style:normal;"><a href="/wiki/Donald_E._Knuth" class="mw-redirect" title="Donald E. Knuth">KNUTH, Donald E</a>. Two notes on notation. <i>The American Mathematical Monthly</i>. 1992, roč. 99, čís. 5, s. 408. <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1992-05_99_5/page/408">Dostupné online</a>. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a href="https://arxiv.org/abs/math/9205211" class="extiw" title="arxiv:math/9205211">math/9205211</a>. (anglicky)</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceperiodika&rft.jtitle=The+American+Mathematical+Monthly&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_american-mathematical-monthly_1992-05_99_5%2Fpage%2F408&rft.atitle=Two+notes+on+notation&rft.date=1992&rft.volume=99&rft.issue=5&rft.pages=408&rft.aulast=Knuth&rft.aufirst=Donald+E"><span style="display:none"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=0^0">WolframAlpha</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=8" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=8" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Odmocnina" title="Odmocnina">Odmocnina</a></li> <li><a href="/wiki/Logaritmus" title="Logaritmus">Logaritmus</a></li> <li><a href="/wiki/Mocninn%C3%A1_funkce" title="Mocninná funkce">Mocninná funkce</a></li> <li><a href="/wiki/Exponenci%C3%A1ln%C3%AD_funkce" title="Exponenciální funkce">Exponenciální funkce</a></li> <li><a href="/wiki/Geometrick%C3%A1_%C5%99ada" class="mw-redirect" title="Geometrická řada">Geometrická řada</a></li> <li><a href="/wiki/Ko%C5%99en_(matematika)" title="Kořen (matematika)">Kořen (matematika)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&veaction=edit&section=9" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Umoc%C5%88ov%C3%A1n%C3%AD&action=edit&section=9" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span class="sisterproject_text_target"><a href="https://commons.wikimedia.org/wiki/Category:Exponentiation" class="extiw" title="c:Category:Exponentiation">umocňování</a></span> na Wikimedia Commons</span></span></span><i> </i></li> <li><span class="sisterproject sisterproject-wiktionary"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/24px-Wiktionary-logo-cs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/32px-Wiktionary-logo-cs.svg.png 2x" data-file-width="411" data-file-height="411" /></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Slovníkové heslo </span><span class="sisterproject_text_target"><a href="https://cs.wiktionary.org/wiki/mocnina" class="extiw" title="wikt:mocnina">mocnina</a></span><span class="sisterproject_text_suffix"> ve Wikislovníku</span></span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r23078045">.mw-parser-output .navbox2{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox2 .navbox2{margin-top:0}.mw-parser-output .navbox2+.navbox2{margin-top:-1px}.mw-parser-output .navbox2-inner,.mw-parser-output .navbox2-subgroup{width:100%}.mw-parser-output .navbox2-group,.mw-parser-output .navbox2-title,.mw-parser-output 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<ul><li><span class="nowrap"><a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85046490">sh85046490</a></span></span></li> <li><span class="nowrap"><a href="/wiki/N%C3%A1rodn%C3%AD_knihovna_Izraele" title="Národní knihovna Izraele">NLI</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007562810505171">987007562810505171</a></span></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Citováno z „<a dir="ltr" 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