{"id":232737,"date":"2025-06-12T06:14:49","date_gmt":"2025-06-12T06:14:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=232737"},"modified":"2025-06-12T06:14:51","modified_gmt":"2025-06-12T06:14:51","slug":"the-cube-root-of-x-times-the-square-root-of-x-can-be-rewritten-as-x-to-the","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/the-cube-root-of-x-times-the-square-root-of-x-can-be-rewritten-as-x-to-the\/","title":{"rendered":"The cube root of x times the square root of x can be rewritten as x to the"},"content":{"rendered":"\n<p>The cube root of x times the square root of x can be rewritten as x to the<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The cube root of xxx times the square root of xxx can be rewritten as:x3\u22c5x=x13\u22c5x12=x56\\sqrt[3]{x} \\cdot \\sqrt{x} = x^{\\frac{1}{3}} \\cdot x^{\\frac{1}{2}} = x^{\\frac{5}{6}}3x\u200b\u22c5x\u200b=x31\u200b\u22c5x21\u200b=x65\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Textbook-Style Explanation<\/h3>\n\n\n\n<p>In algebra, radicals such as square roots and cube roots can be rewritten using rational exponents. This allows us to apply the laws of exponents more easily.<\/p>\n\n\n\n<p>Let\u2019s begin with the original expression:x3\u22c5x\\sqrt[3]{x} \\cdot \\sqrt{x}3x\u200b\u22c5x\u200b<\/p>\n\n\n\n<p>This expression involves two radicals: the cube root of xxx, and the square root of xxx. To simplify, we first rewrite both radicals using exponent notation.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The cube root of xxx is written as x1\/3x^{1\/3}x1\/3<\/li>\n\n\n\n<li>The square root of xxx is written as x1\/2x^{1\/2}x1\/2<\/li>\n<\/ul>\n\n\n\n<p>So the original expression becomes:x1\/3\u22c5x1\/2x^{1\/3} \\cdot x^{1\/2}x1\/3\u22c5x1\/2<\/p>\n\n\n\n<p>Now we apply the <strong>Product of Powers Property<\/strong>, which states that when multiplying two expressions with the same base, we can add the exponents:x1\/3\u22c5x1\/2=x1\/3+1\/2x^{1\/3} \\cdot x^{1\/2} = x^{1\/3 + 1\/2}x1\/3\u22c5x1\/2=x1\/3+1\/2<\/p>\n\n\n\n<p>To add the exponents, we need a common denominator. The least common denominator of 3 and 2 is 6, so we convert both fractions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>13=26\\frac{1}{3} = \\frac{2}{6}31\u200b=62\u200b<\/li>\n\n\n\n<li>12=36\\frac{1}{2} = \\frac{3}{6}21\u200b=63\u200b<\/li>\n<\/ul>\n\n\n\n<p>Now add:x2\/6+3\/6=x5\/6x^{2\/6 + 3\/6} = x^{5\/6}x2\/6+3\/6=x5\/6<\/p>\n\n\n\n<p>So the expression simplifies to:x5\/6x^{5\/6}x5\/6<\/p>\n\n\n\n<p>This simplified form is useful in algebraic manipulation and calculus because rational exponents are easier to differentiate or integrate than radical expressions. Converting between radical form and exponent form is a foundational skill in algebra that helps make more advanced math accessible.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-346.jpeg\" alt=\"\" class=\"wp-image-232738\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The cube root of x times the square root of x can be rewritten as x to the The Correct Answer and Explanation is: The cube root of xxx times the square root of xxx can be rewritten as:x3\u22c5x=x13\u22c5x12=x56\\sqrt[3]{x} \\cdot \\sqrt{x} = x^{\\frac{1}{3}} \\cdot x^{\\frac{1}{2}} = x^{\\frac{5}{6}}3x\u200b\u22c5x\u200b=x31\u200b\u22c5x21\u200b=x65\u200b Textbook-Style Explanation In algebra, radicals such as square [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-232737","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232737","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=232737"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232737\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232737"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232737"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}