{"id":225628,"date":"2025-06-04T10:58:26","date_gmt":"2025-06-04T10:58:26","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=225628"},"modified":"2025-06-04T10:58:28","modified_gmt":"2025-06-04T10:58:28","slug":"the-simplest-form-of-64-729-1-6-is-2-33-24-33-4","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/04\/the-simplest-form-of-64-729-1-6-is-2-33-24-33-4\/","title":{"rendered":"The simplest form of (64\/729) ^-1\/6 is -2\/33\/24\/33\/4"},"content":{"rendered":"\n

The simplest form of (64\/729) ^-1\/6 is -2\/33\/24\/33\/4<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To simplify the expression: (64729)\u221216\\left(\\frac{64}{729}\\right)^{-\\frac{1}{6}}<\/p>\n\n\n\n

\n\n\n\n

\u2705 Step 1: Apply the negative exponent rule<\/strong><\/h3>\n\n\n\n

a\u2212n=1ana^{-n} = \\frac{1}{a^n}<\/p>\n\n\n\n

So: (64729)\u221216=(72964)16\\left(\\frac{64}{729}\\right)^{-\\frac{1}{6}} = \\left(\\frac{729}{64}\\right)^{\\frac{1}{6}}<\/p>\n\n\n\n

\n\n\n\n

\u2705 Step 2: Prime factorization of 729 and 64<\/strong><\/h3>\n\n\n\n
    \n
  • 64=2664 = 2^6<\/li>\n\n\n\n
  • 729=36729 = 3^6<\/li>\n<\/ul>\n\n\n\n

    So: (72964)16=(3626)16=36\u00d71626\u00d716=32\\left(\\frac{729}{64}\\right)^{\\frac{1}{6}} = \\left(\\frac{3^6}{2^6}\\right)^{\\frac{1}{6}} = \\frac{3^{6 \\times \\frac{1}{6}}}{2^{6 \\times \\frac{1}{6}}} = \\frac{3}{2}<\/p>\n\n\n\n

    \n\n\n\n

    \u2705 Final Simplified Answer:<\/strong><\/h3>\n\n\n\n

    (64729)\u221216=32\\left(\\frac{64}{729}\\right)^{-\\frac{1}{6}} = \\frac{3}{2}<\/p>\n\n\n\n

    \n\n\n\n

    \ud83d\udd0d Explanation<\/h2>\n\n\n\n

    The expression (64729)\u22121\/6\\left(\\frac{64}{729}\\right)^{-1\/6} involves two concepts: negative exponents<\/strong> and rational (fractional) exponents<\/strong>.<\/p>\n\n\n\n

    First, recall the negative exponent rule<\/strong>, which tells us that: a\u2212n=1ana^{-n} = \\frac{1}{a^n}<\/p>\n\n\n\n

    So, the negative exponent simply inverts the fraction. Thus, (64729)\u22121\/6\\left(\\frac{64}{729}\\right)^{-1\/6} becomes (72964)1\/6\\left(\\frac{729}{64}\\right)^{1\/6}.<\/p>\n\n\n\n

    Next, we simplify the new base 72964\\frac{729}{64}. To do this, factor both numbers into their prime components:<\/p>\n\n\n\n

      \n
    • 64=2664 = 2^6: this is because 2\u00d72\u00d72\u00d72\u00d72\u00d72=642 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 = 64.<\/li>\n\n\n\n
    • 729=36729 = 3^6: because 3\u00d73\u00d73\u00d73\u00d73\u00d73=7293 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 = 729.<\/li>\n<\/ul>\n\n\n\n

      Therefore: (72964)1\/6=(3626)1\/6\\left(\\frac{729}{64}\\right)^{1\/6} = \\left(\\frac{3^6}{2^6}\\right)^{1\/6}<\/p>\n\n\n\n

      Using the power of a power rule<\/strong> (am)n=amn(a^m)^n = a^{mn}, apply the exponent: 36\u00d71626\u00d716=3121=32\\frac{3^{6 \\times \\frac{1}{6}}}{2^{6 \\times \\frac{1}{6}}} = \\frac{3^1}{2^1} = \\frac{3}{2}<\/p>\n\n\n\n

      So the simplified result is 32\\frac{3}{2}.<\/p>\n\n\n\n

      This result is a rational number and represents the sixth root of the reciprocal of 64729\\frac{64}{729}. Despite the initial complexity, recognizing perfect sixth powers and using exponent rules allows for quick simplification. The incorrect answers like \u201c-2\/33\/24\/33\/4\u201d appear to be typographical errors or miscalculations. The correct and simplest form is: 32\\boxed{\\frac{3}{2}}<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      The simplest form of (64\/729) ^-1\/6 is -2\/33\/24\/33\/4 The Correct Answer and Explanation is: To simplify the expression: (64729)\u221216\\left(\\frac{64}{729}\\right)^{-\\frac{1}{6}} \u2705 Step 1: Apply the negative exponent rule a\u2212n=1ana^{-n} = \\frac{1}{a^n} So: (64729)\u221216=(72964)16\\left(\\frac{64}{729}\\right)^{-\\frac{1}{6}} = \\left(\\frac{729}{64}\\right)^{\\frac{1}{6}} \u2705 Step 2: Prime factorization of 729 and 64 So: (72964)16=(3626)16=36\u00d71626\u00d716=32\\left(\\frac{729}{64}\\right)^{\\frac{1}{6}} = \\left(\\frac{3^6}{2^6}\\right)^{\\frac{1}{6}} = \\frac{3^{6 \\times \\frac{1}{6}}}{2^{6 \\times \\frac{1}{6}}} = \\frac{3}{2} […]<\/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-225628","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225628","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=225628"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225628\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=225628"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=225628"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=225628"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}