{"id":67315,"date":"2026-10-21T09:49:47","date_gmt":"2026-10-21T06:49:47","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=67315"},"modified":"2026-10-21T09:49:47","modified_gmt":"2026-10-21T06:49:47","slug":"square-root-calculator-simplify-radicals","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/square-root-calculator-simplify-radicals\/","title":{"rendered":"Square Root Calculator \u2013 Simplify Radicals"},"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\">Simplify a Square Root, Step by Step<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"67315\" data-favorite-title=\"Square Root\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/square-root-calculator-simplify-radicals\/\" data-favorite-excerpt=\"Enter any number \u2014 integer, decimal or fraction \u2014 and this square root calculator gives the decimal value, checks whether it is a perfect square, and shows the simplified radical step by step using prime factorization.\" 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\/square-root-calculator-simplify-radicals\/\" method=\"POST\" class=\"calculator\" id=\"calculator-67315\" data-post=\"67315\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Mode<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"mode\"><option value=\"sq\" selected>Square root \u221ax<\/option><option value=\"nth\">n-th root \u207f\u221ax<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Number (x)<\/label><div class=\"term\">x<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"number_x\" id=\"field-2\" aria-describedby=\"calc-info-2\" placeholder=\"e.g. 72, -16, 2.25 or 3\/4\" step=\"any\" value=\"72\"\/><small id=\"calc-info-2\">Integer, decimal (dot or comma) or fraction a\/b. Negative numbers are allowed: \u221a(\u221216) = 4i, while odd roots stay real, e.g. \u221b(\u221227) = \u22123. Numerator and denominator up to 10\u00b9\u00b2 after simplifying.<\/small><\/div><div class=\"field has-term\" id=\"input-3\" hidden><label for=\"field-3\">Root index (n)<\/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=\"root_index_n\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"2\" max=\"12\" value=\"3\"\/><small id=\"calc-info-3\">Only for n-th root mode: a whole number from 2 to 12 \u2014 3 for a cube root, 4 for a fourth root, and so on.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"67315\">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>Root value<\/span><span class=\"result-term\">\u221ax<\/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 class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Perfect square?<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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 class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Simplified radical form<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/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 class=\"field-result is-column\" id=\"output-4\"><span class=\"field-title\"><span>Steps: prime factorization<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/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 class=\"field-result is-column\" id=\"output-5\"><span class=\"field-title\"><span>Perfect squares 1\u201330<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/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=\"67315\" 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\">1<\/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>Square 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{x} = a\\sqrt{b} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(x\\) \u2014 the number under the radical sign, called the radicand<\/li>\n<li>\\(a\\) \u2014 the whole-number coefficient pulled out of the root<\/li>\n<li>\\(b\\) \u2014 the squarefree part that stays inside the radical (no perfect square greater than 1 divides it)<\/li>\n<\/ul>\n<p>To find \\(a\\) and \\(b\\), write the prime factorization \\(x = p_1^{e_1} \\cdot p_2^{e_2} \\cdots\\). Each prime leaves the root with half its exponent and keeps the remainder inside: \\(a = p_1^{\\lfloor e_1\/2 \\rfloor} \\cdot p_2^{\\lfloor e_2\/2 \\rfloor} \\cdots\\) and \\(b = p_1^{e_1 \\bmod 2} \\cdot p_2^{e_2 \\bmod 2} \\cdots\\). For fractions, the same rule applies to the numerator and denominator separately, then the denominator is rationalized.<\/p>\n<p>If \\(b = 1\\), no radical remains and \\(\\sqrt{x} = a\\) is a whole number \u2014 a perfect square. If no perfect square greater than 1 divides \\(x\\), then \\(a = 1\\) and the radical is already in its simplest form.<\/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>Square Root \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> simplify \\(\\sqrt{72}\\), the calculator's default value.<\/p>\n<p>\\[ 72 = 2^3 \\cdot 3^2 \\]<\/p>\n<p>\\[ \\sqrt{72} = \\sqrt{2^3 \\cdot 3^2} = \\sqrt{(2 \\cdot 3)^2 \\cdot 2} = 6\\sqrt{2} \\]<\/p>\n<p>The prime 2 appears three times: one pair leaves the root as a factor of 2, and a single 2 stays inside. The pair of 3s leaves the root as one factor of 3, so the coefficient is \\(2 \\cdot 3 = 6\\).<\/p>\n<p><strong>Result:<\/strong> \\(\\sqrt{72} = 6\\sqrt{2} \\approx 8.48528137423857\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li><strong>Pick the Mode.<\/strong> <em>Square root \u221ax<\/em> is the default. Choose <em>n-th root \u207f\u221ax<\/em> if you need a cube root, fourth root or any other order \u2014 an extra <em>Root index (n)<\/em> field appears for that.<\/li>\n<li><strong>Enter the Number (x).<\/strong> Type an integer (72), a decimal (2.25), or a fraction (3\/4). A leading minus sign is allowed: even roots of negative numbers come back as imaginary, like \\(\\sqrt{-16} = 4i\\), while odd roots stay real, like \\(\\sqrt[3]{-27} = -3\\).<\/li>\n<li><strong>Read the results.<\/strong> <em>Root value<\/em> shows the decimal answer; <em>Perfect square?<\/em> tells you whether the root is a whole (or rational) number; <em>Simplified radical form<\/em> gives the exact answer such as \\(6\\sqrt{2}\\); <em>Steps: prime factorization<\/em> shows the full simplification with every prime exponent; and the <em>Perfect squares 1\u201330<\/em> table helps with hand calculations.<\/li>\n<\/ol>\n<h2>What does it mean to simplify a square root?<\/h2>\n<p>The square root of \\(x\\) is the non-negative number that, multiplied by itself, gives \\(x\\): \\(\\sqrt{x} \\cdot \\sqrt{x} = x\\). For perfect squares such as 25 or 144, the root is a whole number \u2014 5 and 12 respectively. For every other positive number, the root is irrational: its decimal never ends and never repeats, so a calculator can only show an approximation.<\/p>\n<p>Simplifying a radical means rewriting that irrational root in its shortest exact form. The idea is to pull out every perfect square hiding inside the radicand. Since \\(72 = 36 \\cdot 2\\), you get \\(\\sqrt{72} = \\sqrt{36} \\cdot \\sqrt{2} = 6\\sqrt{2}\\). Nothing more can come out, because 2 has no square factor \u2014 the radical is now <em>fully simplified<\/em>. This exact form is preferred in algebra: it is precise, it makes further steps (like adding \\(\\sqrt{8} = 2\\sqrt{2}\\) to \\(6\\sqrt{2}\\)) possible, and it reveals when two apparently different radicals are actually the same number.<\/p>\n<p>The <em>Perfect square?<\/em> line tells you the nature of your answer. If it says yes, the root is rational and the radical disappears entirely. If it says no, the number lies between two consecutive perfect squares \u2014 for example, \\(\\sqrt{40}\\) is between \\(\\sqrt{36} = 6\\) and \\(\\sqrt{49} = 7\\) \u2014 and the simplified radical is the only exact way to write it.<\/p>\n<p>Fractions are handled exactly. The input is first reduced to lowest terms, then the denominator is rationalized: \\(\\sqrt{3\/4} = \\sqrt{3}\/2\\), not a rounded decimal. Decimals up to twelve places are converted to fractions internally, so the simplification stays exact.<\/p>\n<h2>Assumptions, limits and tips<\/h2>\n<ul>\n<li><strong>Principal root.<\/strong> For a positive \\(x\\), the calculator returns the positive (principal) root. Remember that \\(x^2 = 72\\) has <em>two<\/em> solutions: \\(\\pm 6\\sqrt{2}\\).<\/li>\n<li><strong>Negative numbers.<\/strong> An even root of a negative number is imaginary: \\(\\sqrt{-16} = 4i\\). In n-th root mode with an odd index, the answer stays real: \\(\\sqrt[3]{-27} = -3\\).<\/li>\n<li><strong>Size limits.<\/strong> After simplification, the numerator and denominator may each be at most 10<sup>12<\/sup>, with at most 13 significant digits entered \u2014 enough for any school, textbook or competition problem.<\/li>\n<li><strong>Decimal answers<\/strong> are shown with up to 15 significant digits, so the last digit may differ from a value you round differently. Use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/rounding-calculator\/\">Rounding Calculator<\/a> when a specific precision is required.<\/li>\n<li><strong>Simplify by hand:<\/strong> divide the radicand by the largest perfect square factor you can spot from the 1\u201330 table, or check individual divisors with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/factor-calculator-all-factors-of-a-number\/\">Factor Calculator<\/a>.<\/li>\n<li>The step-by-step breakdown relies on prime factorization \u2014 see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/prime-factorization-calculator\/\">Prime Factorization Calculator<\/a> for that part in isolation, or the <a href=\"https:\/\/wpcalc.com\/en\/nth-root-calculator\/\">Nth Root Calculator<\/a> for roots of any order in full detail.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter any number \u2014 integer, decimal or fraction \u2014 and this square root calculator gives the decimal value, checks whether it is a perfect square, and shows the simplified radical step by step using prime factorization.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[72],"class_list":["post-67315","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\/67315","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=67315"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67315\/revisions"}],"predecessor-version":[{"id":67316,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67315\/revisions\/67316"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=67315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=67315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=67315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}