{"id":65057,"date":"2025-07-29T10:49:21","date_gmt":"2025-07-29T07:49:21","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65057"},"modified":"2026-09-30T20:42:37","modified_gmt":"2026-09-30T17:42:37","slug":"nth-root","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/nth-root\/","title":{"rendered":"Nth Root Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Nth Root Solver<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65057\" data-favorite-title=\"Nth Root\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/nth-root\/\" data-favorite-excerpt=\"Find the nth root of any real number: enter the number and the root degree (2 for square root, 3 for cube root, or any degree up to 100). The calculator returns the exact decimal value instantly.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-arithmetic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/nth-root\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65057\" data-post=\"65057\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Number<\/label><div class=\"term\">A<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"number\" id=\"field-1\" aria-describedby=\"calc-info-1\" min=\"-1000000\" max=\"1000000\" step=\"any\" value=\"8\"\/><small id=\"calc-info-1\">Enter the number you want to extract the root from<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Degree of the root<\/label><div class=\"term\">N<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"degree_of_the_root\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1\" max=\"100\" value=\"3\"\/><small id=\"calc-info-2\">Enter the root degree (e.g. 2 for square root, 3 for cube root)<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65057\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Nth Root<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65057\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Nth Root Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\sqrt[n]{x} = x^{\\frac{1}{n}} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(x\\) \u2014 the number (radicand) you want to take the root of<\/li>\n<li>\\(n\\) \u2014 the degree of the root (a positive integer: 2 for a square root, 3 for a cube root, and so on)<\/li>\n<li>\\(\\sqrt[n]{x}\\) \u2014 the nth root of \\(x\\): the number \\(y\\) that satisfies \\(y^n = x\\)<\/li>\n<\/ul>\n<p>The formula expresses the nth root as a fractional exponent. Taking the nth root of a number is exactly the same as raising that number to the power \\(1\/n\\), so the two notations are interchangeable.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Nth Root \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the cube root of 8 (the calculator's default values: Number = 8, Degree of the root = 3).<\/p>\n<p>\\[ \\sqrt[3]{8} = 8^{\\frac{1}{3}} = 2 \\]<\/p>\n<p><strong>Check:<\/strong> \\(2^3 = 2 \\times 2 \\times 2 = 8\\), so the answer is correct.<\/p>\n<p><strong>Result:<\/strong> the cube root of 8 is <strong>2<\/strong>.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Nth Root Calculator<\/h2>\n<ol>\n<li>Type the number you want to extract the root from into the <strong>Number<\/strong> field (for example, 8).<\/li>\n<li>Enter the root degree in the <strong>Degree of the root<\/strong> field \u2014 use 2 for a square root, 3 for a cube root, or any integer from 1 to 100.<\/li>\n<li>Read the result in the <strong>Nth Root<\/strong> field. The answer is rounded to 6 decimal places.<\/li>\n<\/ol>\n<p>To verify a result, raise it back to the power \\(n\\) with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/multiplication-calculator\/\">multiplication calculator<\/a> \u2014 you should get your original number back.<\/p>\n<h2>What Is an Nth Root?<\/h2>\n<p>The nth root generalizes the familiar square root and cube root to any degree. The nth root of a number \\(x\\) is the value \\(y\\) that, when multiplied by itself \\(n\\) times, gives \\(x\\). In symbols: \\(y = \\sqrt[n]{x}\\) means \\(y^n = x\\). The square root (\\(n=2\\)) and cube root (\\(n=3\\)) are simply the two most common special cases.<\/p>\n<p>The key idea behind the formula is the equivalence between roots and fractional exponents. Because \\(\\sqrt[n]{x} = x^{1\/n}\\), any root operation can be rewritten as exponentiation. This makes roots easy to evaluate with a calculator and convenient to combine with other exponent rules, for example \\(\\sqrt[n]{x^m} = x^{m\/n}\\).<\/p>\n<p>Nth roots appear everywhere in practice. In geometry, the side length of a cube with volume \\(V\\) is \\(\\sqrt[3]{V}\\), and the side of a square with area \\(A\\) is \\(\\sqrt{A}\\). In finance, the compound annual growth rate is computed as an nth root: over \\(n\\) years, the average growth factor is \\(\\sqrt[n]{\\text{final}\/\\text{initial}}\\). In statistics, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/geometric-mean\/\">geometric mean<\/a> of \\(n\\) values is the nth root of their product, and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-mean-calculator-root-mean-square\/\">quadratic mean (root mean square)<\/a> finishes with a square root.<\/p>\n<p>When interpreting the result, remember that for a positive radicand and an odd degree there is exactly one real root, and for an even degree the principal (positive) root is the standard answer. For example, both \\(2\\) and \\(-2\\) squared give 4, but \\(\\sqrt{4} = 2\\) by convention.<\/p>\n<h2>Common Roots Reference<\/h2>\n<table>\n<thead>\n<tr>\n<th>Degree \\(n\\)<\/th>\n<th>Name<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>Square root<\/td>\n<td>\\(\\sqrt{25} = 5\\)<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>Cube root<\/td>\n<td>\\(\\sqrt[3]{27} = 3\\)<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>Fourth root<\/td>\n<td>\\(\\sqrt[4]{16} = 2\\)<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>Fifth root<\/td>\n<td>\\(\\sqrt[5]{32} = 2\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Degree range:<\/strong> the calculator accepts root degrees from 1 to 100. A degree of 1 returns the number itself, since \\(x^1 = x\\).<\/li>\n<li><strong>Negative numbers:<\/strong> odd roots of negative numbers are well defined (for example, \\(\\sqrt[3]{-8} = -2\\)), and the calculator handles them correctly. Even roots of negative numbers, however, have no real result \u2014 \\(\\sqrt{-4}\\) is not a real number \u2014 so use negative inputs only with odd degrees.<\/li>\n<li><strong>Rounding:<\/strong> results are rounded to 6 decimal places. Many roots, such as \\(\\sqrt{2} \\approx 1.414214\\), are irrational, so the displayed value is an approximation.<\/li>\n<li><strong>Non-integer degrees:<\/strong> the calculator expects an integer degree. For expressions like \\(x^{2.5}\\), rewrite them using exponent rules (e.g. \\(x^{2.5} = x^2 \\sqrt{x}\\)) before calculating.<\/li>\n<li><strong>Sanity check:<\/strong> after any calculation, raise the result to the power \\(n\\) \u2014 you should recover the original number (up to rounding).<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Find the nth root of any real number: enter the number and the root degree (2 for square root, 3 for cube root, or any degree up to 100). The calculator returns the exact decimal value instantly.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[72],"class_list":["post-65057","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-arithmetic-calculators"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65057"}],"version-history":[{"count":5,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65057\/revisions"}],"predecessor-version":[{"id":66271,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65057\/revisions\/66271"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65057"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65057"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}