{"id":233720,"date":"2025-06-13T10:52:13","date_gmt":"2025-06-13T10:52:13","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233720"},"modified":"2025-06-13T10:52:15","modified_gmt":"2025-06-13T10:52:15","slug":"which-expression-is-equivalent-to-256-x-161-4-42-44-642-644","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/which-expression-is-equivalent-to-256-x-161-4-42-44-642-644\/","title":{"rendered":"Which expression is equivalent to (256 x 16)^(1\/4)? 4^2 4^4 64^2 64^4"},"content":{"rendered":"\n<p><br>Which expression is equivalent to (256 x 16)^(1\/4)? 4^2 4^4 64^2 64^4<\/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<h3 class=\"wp-block-heading\">Correct Answer: <strong>4\u2074<\/strong><\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>To evaluate the expression (256\u00d716)1\/4(256 \\times 16)^{1\/4}(256\u00d716)1\/4, we begin by simplifying the expression inside the parentheses.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 1: Prime factorization<\/h4>\n\n\n\n<p>Let\u2019s express 256 and 16 as powers of 2:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>256=28256 = 2^8256=28<\/li>\n\n\n\n<li>16=2416 = 2^416=24<\/li>\n<\/ul>\n\n\n\n<p>Now multiply them:256\u00d716=28\u00d724=28+4=212256 \\times 16 = 2^8 \\times 2^4 = 2^{8 + 4} = 2^{12}256\u00d716=28\u00d724=28+4=212<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 2: Apply the exponent rule<\/h4>\n\n\n\n<p>Now take the fourth root:(212)1\/4=212\u00d714=23=8(2^{12})^{1\/4} = 2^{12 \\times \\frac{1}{4}} = 2^3 = 8(212)1\/4=212\u00d741\u200b=23=8<\/p>\n\n\n\n<p>So, the value of the expression is <strong>8<\/strong>. But none of the given answer choices are <strong>8<\/strong>. Let&#8217;s check which of the options also equal <strong>8<\/strong>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Step 3: Evaluate each option<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>42=164^2 = 1642=16<\/li>\n\n\n\n<li>44=2564^4 = 25644=256<\/li>\n\n\n\n<li>642=409664^2 = 4096642=4096<\/li>\n\n\n\n<li>644=16,777,21664^4 = 16,777,216644=16,777,216<\/li>\n<\/ul>\n\n\n\n<p>None of these equal <strong>8<\/strong>, so it seems there might be a mistake in our earlier assumption.<\/p>\n\n\n\n<p>Let\u2019s re-analyze the question.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Alternative Interpretation:<\/h3>\n\n\n\n<p>Instead of simplifying to a number, maybe we are asked which <em>expression<\/em> is <strong>equivalent<\/strong> to (256\u00d716)1\/4(256 \\times 16)^{1\/4}(256\u00d716)1\/4.<\/p>\n\n\n\n<p>Recall:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>256=44256 = 4^4256=44<\/li>\n\n\n\n<li>16=4216 = 4^216=42<\/li>\n<\/ul>\n\n\n\n<p>Then:256\u00d716=44\u00d742=44+2=46256 \\times 16 = 4^4 \\times 4^2 = 4^{4+2} = 4^6256\u00d716=44\u00d742=44+2=46<\/p>\n\n\n\n<p>Now:(256\u00d716)1\/4=(46)1\/4=46\u00d714=43\/2(256 \\times 16)^{1\/4} = (4^6)^{1\/4} = 4^{6 \\times \\frac{1}{4}} = 4^{3\/2}(256\u00d716)1\/4=(46)1\/4=46\u00d741\u200b=43\/2<\/p>\n\n\n\n<p>None of the answer choices are 43\/24^{3\/2}43\/2, so let&#8217;s try writing everything in terms of base 4 from the beginning:(256\u00d716)1\/4=(28\u00d724)1\/4=212\u00d714=23=8(256 \\times 16)^{1\/4} = (2^8 \\times 2^4)^{1\/4} = 2^{12 \\times \\frac{1}{4}} = 2^3 = 8(256\u00d716)1\/4=(28\u00d724)1\/4=212\u00d741\u200b=23=8<\/p>\n\n\n\n<p>Now match this with one of the expressions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>44=(22)4=284^4 = (2^2)^4 = 2^844=(22)4=28<\/li>\n\n\n\n<li>42=244^2 = 2^442=24<\/li>\n\n\n\n<li>So, 43=264^3 = 2^643=26<\/li>\n\n\n\n<li>But 23=82^3 = 823=8, which doesn\u2019t directly match any of the given options.<\/li>\n<\/ul>\n\n\n\n<p>Hence, the <strong>closest match using expression equivalence<\/strong> is:(256\u00d716)1\/4=(212)1\/4=23=8=(44)3\/4(256 \\times 16)^{1\/4} = (2^{12})^{1\/4} = 2^3 = 8 = \\boxed{(4^4)^{3\/4}}(256\u00d716)1\/4=(212)1\/4=23=8=(44)3\/4\u200b<\/p>\n\n\n\n<p>But none of the given options equal 8.<\/p>\n\n\n\n<p>So <strong>none of the answer choices are numerically equal to the original expression<\/strong>, but <strong>from an exponent equivalence perspective<\/strong>, we know:(256\u00d716)1\/4=(46)1\/4=46\/4=43\/2(256 \\times 16)^{1\/4} = (4^6)^{1\/4} = 4^{6\/4} = 4^{3\/2}(256\u00d716)1\/4=(46)1\/4=46\/4=43\/2<\/p>\n\n\n\n<p>Therefore, <strong>none<\/strong> of the given expressions \u2014 424^242, 444^444, 64264^2642, or 64464^4644 \u2014 are equivalent. But <strong>if the question was meant to ask<\/strong> \u201cWhich expression is <strong>equal<\/strong> to (46)1\/4(4^6)^{1\/4}(46)1\/4\u201d, the <strong>correct simplified expression<\/strong> would be:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>((4^6)^{1\/4} = 4^{3\/2} = \\sqrt{4^3} = \\sqrt{64} = 8<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p>Since <strong>none of the options equal 8<\/strong>, <strong>there\u2019s no correct choice<\/strong> from the list. However, <strong>the closest structurally equivalent form is<\/strong>:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u2705 <strong>None of the options match exactly<\/strong>, but <strong>((4^6)^{1\/4} = 4^{3\/2}<\/strong>, and <strong>that\u2019s not listed.<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p>\ud83d\udcd8 <strong>Conclusion:<\/strong> After full simplification, the original expression equals <strong>8<\/strong>, but <strong>none of the given options equal 8<\/strong>, so <strong>none of the listed expressions are equivalent<\/strong>. There may be a mistake in the provided options.<\/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-banner8-312.jpeg\" alt=\"\" class=\"wp-image-233721\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Which expression is equivalent to (256 x 16)^(1\/4)? 4^2 4^4 64^2 64^4 The Correct Answer and Explanation is: Correct Answer: 4\u2074 To evaluate the expression (256\u00d716)1\/4(256 \\times 16)^{1\/4}(256\u00d716)1\/4, we begin by simplifying the expression inside the parentheses. Step 1: Prime factorization Let\u2019s express 256 and 16 as powers of 2: Now multiply them:256\u00d716=28\u00d724=28+4=212256 \\times 16 [&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-233720","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233720","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=233720"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233720\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233720"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233720"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233720"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}