{"id":61464,"date":"2021-02-02T16:07:34","date_gmt":"2021-02-02T14:07:34","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61464"},"modified":"2026-09-30T20:50:54","modified_gmt":"2026-09-30T17:50:54","slug":"sum-two-cubes","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/sum-two-cubes\/","title":{"rendered":"Sum of Two Cubes 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\">Find the Sum of Two Cubes<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61464\" data-favorite-title=\"Sum of Two Cubes\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-two-cubes\/\" data-favorite-excerpt=\"Instantly compute a\u00b3 + b\u00b3 for any two numbers. Enter the values of a and b, and the calculator shows each cube and their sum, with the factoring identity a\u00b3 + b\u00b3 = (a + b)(a\u00b2 \u2212 ab + b\u00b2).\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\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\/sum-two-cubes\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61464\" data-post=\"61464\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">a<\/label><input type=\"number\" name=\"a\" id=\"field-1\" placeholder=\"2\" step=\"1\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">b<\/label><input type=\"number\" name=\"b\" id=\"field-2\" placeholder=\"3\" step=\"1\" value=\"3\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61464\">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>a\u00b3<\/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>b\u00b3<\/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\" id=\"output-3\"><span class=\"field-title\"><span>Sum of cubes<\/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><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=\"61464\" 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>Sum of Two Cubes 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\\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a\\) \u2014 the first number (or term)<\/li>\n<li>\\(b\\) \u2014 the second number (or term)<\/li>\n<li>\\(a^3\\) \u2014 the cube of \\(a\\), i.e. \\(a \\cdot a \\cdot a\\)<\/li>\n<li>\\(b^3\\) \u2014 the cube of \\(b\\), i.e. \\(b \\cdot b \\cdot b\\)<\/li>\n<\/ul>\n<p>On the left side, the formula gives the plain sum of the two cubes; on the right side, it shows the same quantity factored into a binomial \\((a + b)\\) and a trinomial \\((a^2 - ab + b^2)\\). Both sides are always equal, which is what makes this identity so useful for simplifying and factoring polynomial expressions.<\/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>Sum of Two Cubes \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the sum of the cubes of \\(a = 2\\) and \\(b = 3\\).<\/p>\n<p>\\[ a^3 = 2^3 = 2 \\cdot 2 \\cdot 2 = 8 \\]<\/p>\n<p>\\[ b^3 = 3^3 = 3 \\cdot 3 \\cdot 3 = 27 \\]<\/p>\n<p>\\[ a^3 + b^3 = 8 + 27 = 35 \\]<\/p>\n<p><strong>Result:<\/strong> \\(a^3 = 8\\), \\(b^3 = 27\\), and the sum of the cubes is \\(35\\).<\/p>\n<p>You can verify this with the factored form as well:<\/p>\n<p>\\[ (a + b)(a^2 - ab + b^2) = (2 + 3)(4 - 6 + 9) = 5 \\cdot 7 = 35 \\]<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the value of <strong>a<\/strong> in the first field (the default is 2).<\/li>\n<li>Enter the value of <strong>b<\/strong> in the second field (the default is 3).<\/li>\n<li>The results appear instantly: <strong>a\u00b3<\/strong>, <strong>b\u00b3<\/strong>, and the <strong>Sum of cubes<\/strong> \\(a^3 + b^3\\).<\/li>\n<\/ol>\n<p>The calculator accepts decimal and negative numbers, not just integers, so you can use it for any real values of \\(a\\) and \\(b\\).<\/p>\n<h2>What Is the Sum of Two Cubes?<\/h2>\n<p>A <em>cube<\/em> of a number is that number multiplied by itself twice: \\(a^3 = a \\cdot a \\cdot a\\). The <em>sum of two cubes<\/em> is simply \\(a^3 + b^3\\) \u2014 the result of cubing each number separately and then adding the results. For example, with \\(a = 2\\) and \\(b = 3\\), you get \\(8 + 27 = 35\\).<\/p>\n<p>Beyond straightforward arithmetic, the expression \\(a^3 + b^3\\) has a special place in algebra because it can always be rewritten using the identity \\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\\). This turns a sum of two cubes into a product of a binomial and a trinomial \u2014 a process called <em>factoring<\/em>. For instance, \\(x^3 + 27\\) can be seen as \\(x^3 + 3^3\\) and factored as \\((x + 3)(x^2 - 3x + 9)\\).<\/p>\n<p>Why does this matter? Factoring is one of the main tools for solving polynomial equations, simplifying fractions with polynomial numerators and denominators, and finding roots of functions. Recognizing a sum-of-cubes pattern lets you break a cubic expression into simpler pieces. Note that the trinomial factor \\((a^2 - ab + b^2)\\) cannot be factored further using real numbers \u2014 this is where the factored form stops over the reals, although it does factor further using <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-number-calculator\/\">complex numbers<\/a>.<\/p>\n<p>The sum of cubes is closely related to the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/difference-of-cubes\/\">difference of two cubes<\/a>, \\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\\). The two identities look similar, but the signs differ in important ways: in the sum of cubes the binomial has a plus and the middle term of the trinomial is negative, while in the difference of cubes it is the opposite. Mixing up these signs is one of the most common algebra mistakes.<\/p>\n<h2>Tips, Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Works for any real numbers.<\/strong> The values of \\(a\\) and \\(b\\) do not have to be positive or whole. Negative bases work fine: \\((-2)^3 + 3^3 = -8 + 27 = 19\\). Be careful with negative inputs \u2014 cubing a negative number gives a negative result.<\/li>\n<li><strong>Terms must be perfect cubes to factor.<\/strong> The factored identity applies only when each term is a perfect cube (or can be written as one, e.g. \\(64x^6 = (4x^2)^3\\)). If you are factoring an expression, first rewrite each term as something cubed.<\/li>\n<li><strong>Order does not matter for the sum.<\/strong> Since addition is commutative, \\(a^3 + b^3 = b^3 + a^3\\); swapping the two inputs leaves the result unchanged.<\/li>\n<li><strong>Check your factoring by expanding.<\/strong> Multiplying the factored form back out with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/foil-method\/\">FOIL method<\/a> or a general <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/multiplying-binomials\/\">binomial multiplication<\/a> should return the original expression \u2014 a quick way to catch sign errors.<\/li>\n<li><strong>Useful for solving cubic equations.<\/strong> Once an expression like \\(x^3 + 8 = 0\\) is factored as \\((x + 2)(x^2 - 2x + 4) = 0\\), you can find its roots by setting each factor to zero. The <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/cubic-equation\/\">cubic equation calculator<\/a> handles the general case.<\/li>\n<\/ul>\n<p>Historically, sums of cubes also appear in number theory \u2014 for example, in the question of which integers can be written as a sum of two cubes in more than one way (the famous Hardy\u2013Ramanujan number \\(1729 = 1^3 + 12^3 = 9^3 + 10^3\\)). But for everyday algebra homework, this calculator and the identity above cover everything you need.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Instantly compute a\u00b3 + b\u00b3 for any two numbers. Enter the values of a and b, and the calculator shows each cube and their sum, with the factoring identity a\u00b3 + b\u00b3 = (a + b)(a\u00b2 \u2212 ab + b\u00b2).<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-61464","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61464","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\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61464"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61464\/revisions"}],"predecessor-version":[{"id":66295,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61464\/revisions\/66295"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61464"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}