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[o]" accesskey="o" class=""><span>Увійти</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Більше опцій" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Особисті інструменти" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Особисті інструменти</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Меню користувача" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_uk.wikibooks.org&uselang=uk"><span>Пожертвувати</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D1%82%D0%B2%D0%BE%D1%80%D0%B8%D1%82%D0%B8_%D0%BE%D0%B1%D0%BB%D1%96%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Пропонуємо створити обліковий запис і увійти в систему; однак, це не обов'язково"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Створити обліковий запис</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D1%85%D1%96%D0%B4&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Заохочуємо Вас увійти в систему, але це необов'язково. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Увійти</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"> Сторінки для редакторів, які не ввійшли в систему <a href="/wiki/%D0%94%D0%BE%D0%B2%D1%96%D0%B4%D0%BA%D0%B0:%D0%92%D1%81%D1%82%D1%83%D0%BF" aria-label="Дізнатися більше про редагування"><span>дізнатися більше</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/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D1%96%D0%B9_%D0%B2%D0%BD%D0%B5%D1%81%D0%BE%D0%BA" title="Список редагувань, зроблених з цієї IP-адреси [y]" accesskey="y"><span>Внесок</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D0%BE%D1%94_%D0%BE%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Обговорення редагувань з цієї IP-адреси [n]" accesskey="n"><span>Обговорення</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="Сайт"> <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="Зміст" 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">Зміст</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">сховати</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">Вступ</div> </a> </li> <li id="toc-Арифметичний_корінь" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Арифметичний_корінь"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Арифметичний корінь</span> </div> </a> <ul id="toc-Арифметичний_корінь-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="Зміст" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Сховати/показати зміст" > <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">Сховати/показати зміст</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">Основні числові системи/Ірраціональні числа</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="Ця стаття існує лише цією мовою. Додайте статтю для інших мов" > <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-0" 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">Додати мови</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> <div class="after-portlet after-portlet-lang"><span class="uls-after-portlet-link"></span><span class="wb-langlinks-add wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q607728#sitelinks-wikibooks" title="Додати міжмовні посилання" class="wbc-editpage">Додати посилання</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Простори назв"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Вміст статті [c]" accesskey="c"><span>Сторінка</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F:%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&redlink=1" rel="discussion" class="new" title="Обговорення сторінки (такої сторінки не існує) [t]" accesskey="t"><span>Обговорення</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Змінити варіант мови" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">українська</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Перегляди"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0"><span>Читати</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit" title="Редагувати цю сторінку [e]" accesskey="e"><span>Редагувати</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=history" title="Журнал змін сторінки [h]" accesskey="h"><span>Переглянути історію</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Інструменти сторінки"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Інструменти" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Інструменти</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Інструменти</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">сховати</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Більше опцій" > <div class="vector-menu-heading"> Дії </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0"><span>Читати</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit" title="Редагувати цю сторінку [e]" accesskey="e"><span>Редагувати</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=history"><span>Переглянути історію</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Загальний </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D1%81%D0%B8%D0%BB%D0%B0%D0%BD%D0%BD%D1%8F_%D1%81%D1%8E%D0%B4%D0%B8/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Перелік усіх сторінок, які посилаються на цю сторінку [j]" accesskey="j"><span>Посилання сюди</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B2%27%D1%8F%D0%B7%D0%B0%D0%BD%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" rel="nofollow" title="Останні зміни на сторінках, на які посилається ця сторінка [k]" accesskey="k"><span>Пов'язані редагування</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=uk" title="Завантажити файли [u]" accesskey="u"><span>Завантажити файл</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8" title="Перелік спеціальних сторінок [q]" accesskey="q"><span>Спеціальні сторінки</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&oldid=36333" title="Постійне посилання на цю версію цієї сторінки"><span>Постійне посилання</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=info" title="Додаткові відомості про цю сторінку"><span>Інформація про сторінку</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A6%D0%B8%D1%82%D0%B0%D1%82%D0%B0&page=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&id=36333&wpFormIdentifier=titleform" title="Інформація про те, як цитувати цю сторінку"><span>Цитувати сторінку</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:UrlShortener&url=https%3A%2F%2Fuk.wikibooks.org%2Fwiki%2F%25D0%259E%25D1%2581%25D0%25BD%25D0%25BE%25D0%25B2%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE%25D0%25B2%25D1%2596_%25D1%2581%25D0%25B8%25D1%2581%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B8%2F%25D0%2586%25D1%2580%25D1%2580%25D0%25B0%25D1%2586%25D1%2596%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25B0"><span>Отримати вкорочену URL-адресу</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:QrCode&url=https%3A%2F%2Fuk.wikibooks.org%2Fwiki%2F%25D0%259E%25D1%2581%25D0%25BD%25D0%25BE%25D0%25B2%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE%25D0%25B2%25D1%2596_%25D1%2581%25D0%25B8%25D1%2581%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B8%2F%25D0%2586%25D1%2580%25D1%2580%25D0%25B0%25D1%2586%25D1%2596%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25B0"><span>Завантажити QR-код</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Друк/експорт </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0&bookcmd=book_creator&referer=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0"><span>Створити книгу</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:DownloadAsPdf&page=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=show-download-screen"><span>Завантажити як PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&printable=yes" title="Версія цієї сторінки для друку [p]" accesskey="p"><span>Версія до друку</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> В інших проєктах </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Irrational_numbers" hreflang="en"><span>Вікісховище</span></a></li><li class="wb-otherproject-link wb-otherproject-wikipedia mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" hreflang="uk"><span>Вікіпедія</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://uk.wikiquote.org/wiki/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" hreflang="uk"><span>Вікіцитати</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/Q607728" title="Посилання на пов’язаний елемент сховища даних [g]" accesskey="g"><span>Елемент Вікіданих</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="Інструменти сторінки"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Зовнішній вигляд"> <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">Зовнішній вигляд</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">сховати</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">Матеріал з Вікіпідручника</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="subpages">< <bdi dir="ltr"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8" title="Основні числові системи">Основні числові системи</a></bdi></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="uk" dir="ltr"><div class="infobox sisterproject"><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Wikipedia-logo-v2.svg" class="mw-file-description" title="Логотип Вікіпедії"><img alt="Логотип Вікіпедії" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/50px-Wikipedia-logo-v2.svg.png" decoding="async" width="50" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/75px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/100px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></a><figcaption>Логотип Вікіпедії</figcaption></figure> <a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F" class="mw-redirect" title="Вікіпедія">Вікіпедія</a> має пов'язану з цією темою інформацію на сторінці <i><b><a href="https://uk.wikipedia.org/wiki/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" class="extiw" title="w:Ірраціональні числа">Ірраціональні числа</a></b></i> </div> <p>Найновіші відкриття встановили, що ще давні вавилоняни знали про існування ірраціональностей. А у "Началах" Евкліда подається доказ їх існування. Цей доказ виходить з такої основної властивості сумірних й несумірних величин: сумірні величини відносяться між собою, як декотрі (раціональні) числа, а несумірні величини не можуть відноситися між собою як числа. Справді, коли величини <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> сумірні, то вони мають спільну міру. Нехай ця міра міститься у першій величині три рази, а у другій п'ять, тоді <span class="mwe-math-element"><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}{b}}=3:5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>:</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}=3:5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbd18b65e75d45e72fd3ab2484afb2e64ae292e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.073ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}=3:5.}"></span> </p><p>Якщо б несумірні величини відносилися як числа, то це означало б, що вони мають спільну мір. Евклід у дев'ятій теоремі книги встановлює таку умову несумірних відрізків: "Квадрати, які відносяться між собою, як квадратні числа, мають сторони по довжині сумірні, а квадрати, які не відносяться між собою, як декотрі числа, мають сторони за довжиною несумірні". </p><p>Припустімо, що діагональ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> та сторона <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> квадрата є сумірними, тобто <span class="mwe-math-element"><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 {d}{a}}={\frac {p}{q}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{a}}={\frac {p}{q}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5a73c9ceaf6fd71af8ad647552ae1324e77548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.817ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{a}}={\frac {p}{q}},}"></span> де <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> є взаємно простими числами. Тоді <span class="mwe-math-element"><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 {d^{2}}{a^{2}}}={\frac {p^{2}}{q^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{a^{2}}}={\frac {p^{2}}{q^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3e298ec529fb54b8607694096cab2dee703fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.925ex; height:6.343ex;" alt="{\displaystyle {\frac {d^{2}}{a^{2}}}={\frac {p^{2}}{q^{2}}},}"></span> але <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{2}=2a^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2}=2a^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18fc2726c09737a112056f7e382b6f777214995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.464ex; height:3.009ex;" alt="{\displaystyle d^{2}=2a^{2},}"></span> тоді <span class="mwe-math-element"><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 {2a^{2}}{a^{2}}}={\frac {p^{2}}{q^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2a^{2}}{a^{2}}}={\frac {p^{2}}{q^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213b74736e7160ae1d0ef6bd757f761f31aa567e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.088ex; height:6.343ex;" alt="{\displaystyle {\frac {2a^{2}}{a^{2}}}={\frac {p^{2}}{q^{2}}},}"></span> тобто число <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> є парним. Нехай <span class="mwe-math-element"><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=2k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2k,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d2b70c3f90cdbca709be6af2a75fd1bb558e7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.378ex; height:2.509ex;" alt="{\displaystyle p=2k,}"></span> тоді <span class="mwe-math-element"><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^{2}=4k^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}=4k^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb33d3627c6498c6659b5877ecdf97c583464c6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle p^{2}=4k^{2},}"></span> або <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2q^{2}=4k^{2},\,\,q^{2}=2k^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2q^{2}=4k^{2},\,\,q^{2}=2k^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ef279db8d7e39a094d6ec17e4ec6fb7ed726b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.938ex; height:3.009ex;" alt="{\displaystyle 2q^{2}=4k^{2},\,\,q^{2}=2k^{2},}"></span> тобто <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024d4dbdf3feb09055609f33baa8a7ae23aef1d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:3.009ex;" alt="{\displaystyle q^{2}}"></span> та <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> числа парні, що суперечить умові про те, що числа <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> є взаємно простими. Звідси висновок: відрізки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> є несумірними. </p> <div class="mw-heading mw-heading5"><h5 id="Арифметичний_корінь"><span id=".D0.90.D1.80.D0.B8.D1.84.D0.BC.D0.B5.D1.82.D0.B8.D1.87.D0.BD.D0.B8.D0.B9_.D0.BA.D0.BE.D1.80.D1.96.D0.BD.D1.8C"></span>Арифметичний корінь</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=1" title="Редагувати розділ: Арифметичний корінь"><span>ред.</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Арифметичний корінь - невід'ємне значення кореня парного або непарного степеня з невід'ємного числа. Наприклад, коренем з числа <span class="mwe-math-element"><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> <mo>,</mo> </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/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}"></span> формально <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e20cbaee2c94e045df8784f4c398de5862e53f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.813ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}},}"></span> називається таке число <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, квадрат якого дорівнює <span class="mwe-math-element"><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> <mo>,</mo> </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/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}"></span> тобто <span class="mwe-math-element"><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^{2}=a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}=a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be306271a75c1360f4b4600f5f70e753edd197c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.027ex; height:2.676ex;" alt="{\displaystyle b^{2}=a.}"></span> Наприклад, для числа 25 квадратними коренями є числа <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -5,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>5</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -5,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22bcf948d3a780eabfd4ae0e42332cf519037fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.617ex; height:2.509ex;" alt="{\displaystyle -5,}"></span> бо <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{2}=25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{2}=25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c64271a5b077b39e16fb8c91d871a8c9388ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.676ex;" alt="{\displaystyle 5^{2}=25}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-5)^{2}=25.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>25.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-5)^{2}=25.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f34f015abc98b86d161ce0d54a57944262fa126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.904ex; height:3.176ex;" alt="{\displaystyle (-5)^{2}=25.}"></span> Таким чином, для будь-якого додатного числа існує два корені (які є протилежними числами); вважають, що <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {0}}=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>0</mn> </msqrt> </mrow> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {0}}=0;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e25f65033bd9c5f4729d87ebd7d3c7b2b5546e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle {\sqrt {0}}=0;}"></span> для від'ємних чисел квадратних коренів немає. </p><p>Дія вилучення квадратного кореня може розумітися як дія, зворотна до піднесення до степеня (розуміючи під дією вилучення квадратного кореня не збільшення показника степеня, а його зменшення, необхідно визнати, що позначення цієї дії показником <span class="mwe-math-element"><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 {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0aefecf48d43fdedd71e318ae6129bd67be252" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.231ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{n}}}"></span> достатньо ясно виражає поняття "вилучення квадратного кореня"). Знак <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\sqrt[{n}]{}}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow /> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\sqrt[{n}]{}}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e19a6d45b5f5de276663ae79c95ef6f454f69ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.876ex; height:3.509ex;" alt="{\displaystyle [{\sqrt[{n}]{}}],}"></span> уведенй К.Рудольфом, є аналогічним до позначення С.Cтевіна <span class="mwe-math-element"><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 {1}{n}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\frac {1}{n}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1da2c1cd2121c5d35fc599134b1e0978ed9d6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.687ex; height:5.176ex;" alt="{\displaystyle ({\frac {1}{n}}).}"></span> </p><p>При цьому справедливими є наступні рівності: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}=y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}=y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a080f64caf02d5879cfc42fa8773c031a07f68a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}=y,}"></span> якщо <span class="mwe-math-element"><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^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1cfda1f3310a5c649d6847b1b2325968850889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.643ex; height:3.009ex;" alt="{\displaystyle y^{2}=x}"></span> та <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130e8795bc869a5b823133c5a0972693605c00bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 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 ({\sqrt {x}})^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt {x}})^{2}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45a7ff128b02d0222cdfe33f7bdcb57536b9a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.557ex; height:3.343ex;" alt="{\displaystyle ({\sqrt {x}})^{2}=x}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x^{2}}}=|x|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}}}=|x|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56384a43b0ad615d17273a3b7ad2115e159c2cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.076ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2}}}=|x|.}"></span></li></ul> <p>Властивості квадратного кореня: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {xy}}={\sqrt {x}}\cdot {\sqrt {y}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>y</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {xy}}={\sqrt {x}}\cdot {\sqrt {y}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1535739347520eb599ba2a97c82c5690239281" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.202ex; height:3.176ex;" alt="{\displaystyle {\sqrt {xy}}={\sqrt {x}}\cdot {\sqrt {y}},}"></span> якщо <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2608e2b392b079f5b763f27bf52883dbee3b64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}"></span> та <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130e8795bc869a5b823133c5a0972693605c00bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 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 {\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>x</mi> </msqrt> <msqrt> <mi>y</mi> </msqrt> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5c5f6be0109e83bf5350c70bbd1296a761c45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.337ex; height:6.843ex;" alt="{\displaystyle {\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {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 {\sqrt {x^{2n}}}=|x^{n}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2n}}}=|x^{n}|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c338ab1d180553c57b921978cfc4eb528f6a6b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.281ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2n}}}=|x^{n}|.}"></span></li></ul> <p>Корінь степені <span class="mwe-math-element"><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> <mo>,</mo> </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/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"></span> де <span class="mwe-math-element"><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> <mo>,</mo> </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/5b98e213fe7ef48da0be47453bc1bb66f37f4eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.56ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} ,}"></span> з числа <span class="mwe-math-element"><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> є число, яке у степені <span class="mwe-math-element"><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> дорівнює <span class="mwe-math-element"><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> <mo>.</mo> </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/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span> При цьому корінь другого степеня називають квадратним коренем, а корінь третього степеня - кубічним коренем. Зокрема, коренем четвертого степеня з числа <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960615e346e1c003a911f45b1225113ea01b4ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 16}"></span> є числа <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e5b5b462e546b1d3d7e5f9a23efece405b2e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993d53082bc05c266933af9a892e1ce2128547cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 2,}"></span> оскільки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2)^{4}=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2)^{4}=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49bb5babfb5c478d8ab258fd6a6a3e3b725820ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.258ex; height:3.176ex;" alt="{\displaystyle (-2)^{4}=16}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4}=16.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>16.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}=16.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f1cac64b4ddffbf99b7b71cdd76e7d945c6f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.287ex; height:2.676ex;" alt="{\displaystyle 2^{4}=16.}"></span> Кубічний корінь з числа <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -27}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>27</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -27}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adaf5197971ecf199c65e31645c206ce1acef16e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.133ex; height:2.343ex;" alt="{\displaystyle -27}"></span> дорівнює <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d991696f3971b9880a4765191581ca1928d6f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.617ex; height:2.509ex;" alt="{\displaystyle -3,}"></span> оскільки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-3)^{3}=-27.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>27.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-3)^{3}=-27.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e575abaac7008e61352769aed2313f239350b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.712ex; height:3.176ex;" alt="{\displaystyle (-3)^{3}=-27.}"></span> Таким чином, у випадку, якщо <span class="mwe-math-element"><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> є непарним натуральним числом, то існує лише один корінь степеня <span class="mwe-math-element"><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> з довільного числа <span class="mwe-math-element"><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> <mo>.</mo> </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/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span> </p><p>Арифметичним коренем степені <span class="mwe-math-element"><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> з невід'ємного числа <span class="mwe-math-element"><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> є невід'ємне число, яке у степені <span class="mwe-math-element"><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> дорівнює <span class="mwe-math-element"><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> <mo>.</mo> </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/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></span> Арифметичний корінь степені <span class="mwe-math-element"><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> позначають <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8562e64a6bc6e408ddf67f055682c4dc9c9f957f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.912ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}.}"></span> У виппадку, якщо <span class="mwe-math-element"><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> - непарне число, то запис <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}}"></span> використовується й для від'ємних значень <span class="mwe-math-element"><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> (такий корінь не є арифметичним). </p><p>Для арифметичного кореня справджуються наступні рівності: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}=y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}=y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563615e28b5994809e54d5afcaeafe02fbf99dfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}=y,}"></span> якщо <span class="mwe-math-element"><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^{n}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{n}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287a41964d6e61f0251e3e13a9fd032c558e8a41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.676ex;" alt="{\displaystyle y^{n}=x}"></span> та <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130e8795bc869a5b823133c5a0972693605c00bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 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 ({\sqrt[{n}]{x}})^{n}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt[{n}]{x}})^{n}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d725d4bdabe0b2eb8b351f432f74780b341e178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.721ex; height:3.009ex;" alt="{\displaystyle ({\sqrt[{n}]{x}})^{n}=x}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}^{n}=|x|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}^{n}=|x|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d14d928355919b288c73faf9a11087b6cf9bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.853ex; height:3.176ex;" alt="{\displaystyle {\sqrt[{n}]{x}}^{n}=|x|.}"></span></li></ul> <p>Властивості арифметичного кореня: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{xy}}={\sqrt[{n}]{x}}\cdot {\sqrt[{n}]{y}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{xy}}={\sqrt[{n}]{x}}\cdot {\sqrt[{n}]{y}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3726c920d1bda258683031fd18a5109e7be2ba36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.202ex; height:3.176ex;" alt="{\displaystyle {\sqrt[{n}]{xy}}={\sqrt[{n}]{x}}\cdot {\sqrt[{n}]{y}},}"></span> якщо <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2608e2b392b079f5b763f27bf52883dbee3b64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}"></span> та <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130e8795bc869a5b823133c5a0972693605c00bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 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 {\sqrt[{n}]{\frac {x}{y}}}={\frac {\sqrt[{n}]{x}}{\sqrt[{n}]{y}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> <mroot> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{\frac {x}{y}}}={\frac {\sqrt[{n}]{x}}{\sqrt[{n}]{y}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fcc1071a652a2202a6bb8b8366acc83e07825b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.337ex; height:6.843ex;" alt="{\displaystyle {\sqrt[{n}]{\frac {x}{y}}}={\frac {\sqrt[{n}]{x}}{\sqrt[{n}]{y}}},}"></span> якщо <span class="mwe-math-element"><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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2608e2b392b079f5b763f27bf52883dbee3b64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}"></span> та <span class="mwe-math-element"><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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c973e3cbfee5d7ab9ca2348b578b6ec19a8c019a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y>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 {\sqrt[{n}]{x^{m}}}=({\sqrt[{n}]{x}})^{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x^{m}}}=({\sqrt[{n}]{x}})^{m},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f0d8d43d5fa0a34405c109ee177c0f49a0cc59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.436ex; height:3.343ex;" alt="{\displaystyle {\sqrt[{n}]{x^{m}}}=({\sqrt[{n}]{x}})^{m},}"></span> якщо <span class="mwe-math-element"><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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>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 {\sqrt[{m\cdot n}]{x^{m}}}={\sqrt[{n}]{x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>⋅</mo> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{m\cdot n}]{x^{m}}}={\sqrt[{n}]{x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56600a13d432286da69033e4016d70bcf439fb19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.01ex; height:3.343ex;" alt="{\displaystyle {\sqrt[{m\cdot n}]{x^{m}}}={\sqrt[{n}]{x}},}"></span> якщо <span class="mwe-math-element"><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\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f783280c16f4847b23b58cd668cf6f99939c7ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x\geq 0,}"></span> а <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> та <span class="mwe-math-element"><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> є натуральними числами <span class="mwe-math-element"><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>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span> та <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f27527902d05e4c32bcbe28d425d7790f8ae191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>1}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{m}]{\sqrt[{n}]{x}}}={\sqrt[{n}]{\sqrt[{m}]{x}}}={\sqrt[{m\cdot n}]{x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mroot> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>⋅</mo> <mi>n</mi> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{m}]{\sqrt[{n}]{x}}}={\sqrt[{n}]{\sqrt[{m}]{x}}}={\sqrt[{m\cdot n}]{x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4dc11ba9270638911aaf3c4abe71f33a4b3121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.346ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{m}]{\sqrt[{n}]{x}}}={\sqrt[{n}]{\sqrt[{m}]{x}}}={\sqrt[{m\cdot n}]{x}},}"></span> якщо <span class="mwe-math-element"><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\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f783280c16f4847b23b58cd668cf6f99939c7ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x\geq 0,}"></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 m>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f27527902d05e4c32bcbe28d425d7790f8ae191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>1}"></span> та <span class="mwe-math-element"><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>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"></span> є натуральними числами.</li></ul> <p><span id="top-navigation" class="noprint" style="top: 0em; right: 1.5em; position:absolute; border-style:outset; border-width:3px; border-color:green;">← <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Основні числові системи/Раціональні числа">Раціональні числа</a> · <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%94%D1%96%D0%B9%D1%81%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Основні числові системи/Дійсні числа">Дійсні числа</a> →</span></p><div id="bottom-navigation" style="float: none; 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