{"id":266687,"date":"2025-07-23T06:30:44","date_gmt":"2025-07-23T06:30:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=266687"},"modified":"2025-07-23T06:30:46","modified_gmt":"2025-07-23T06:30:46","slug":"how-many-real-sixth-roots-does-64-have","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/how-many-real-sixth-roots-does-64-have\/","title":{"rendered":"How many real sixth roots does 64 have"},"content":{"rendered":"\n

How many real sixth roots does 64 have? 0, or 1, or 2<\/p>\n\n\n\n

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

The number 64 has 2 real sixth roots<\/strong>. Here’s an explanation:<\/p>\n\n\n\n

To find the sixth roots of a number, we want to solve for xxx in the equation x6=64x^6 = 64×6=64. We can express 64 as 64=2664 = 2^664=26, so the equation becomes x6=26x^6 = 2^6×6=26. This gives us the principal root and the other possible roots from the complex plane.<\/p>\n\n\n\n

Step-by-step solution:<\/h3>\n\n\n\n
    \n
  1. Expressing 64 in polar form:<\/strong>
    64 is a positive real number, so we can express it in polar form as 64=64(cos\u20610\u2218+isin\u20610\u2218)64 = 64(\\cos 0^\\circ + i\\sin 0^\\circ)64=64(cos0\u2218+isin0\u2218), or simply 64=64ei0\u221864 = 64e^{i0^\\circ}64=64ei0\u2218.<\/li>\n\n\n\n
  2. Using De Moivre’s Theorem:<\/strong>
    The sixth roots of a complex number are given by the formula: xk=r1\/6(cos\u2061(\u03b8+360\u2218k6)+isin\u2061(\u03b8+360\u2218k6))x_k = r^{1\/6} \\left( \\cos \\left(\\frac{\\theta + 360^\\circ k}{6}\\right) + i \\sin \\left(\\frac{\\theta + 360^\\circ k}{6}\\right) \\right)xk\u200b=r1\/6(cos(6\u03b8+360\u2218k\u200b)+isin(6\u03b8+360\u2218k\u200b)) where rrr is the modulus (in this case, 646464), \u03b8\\theta\u03b8 is the argument (for 646464, it’s 0), and kkk takes integer values from 0 to 5, representing the different roots.<\/li>\n\n\n\n
  3. Calculating the roots:<\/strong>
    The modulus of the root is 641\/6=264^{1\/6} = 2641\/6=2, and the argument of the roots is 360\u2218k6=60\u2218k\\frac{360^\\circ k}{6} = 60^\\circ k6360\u2218k\u200b=60\u2218k, where k=0,1,2,3,4,5k = 0, 1, 2, 3, 4, 5k=0,1,2,3,4,5.\n
      \n
    • For k=0k = 0k=0, the root is 2(cos\u20610\u2218+isin\u20610\u2218)=22(\\cos 0^\\circ + i \\sin 0^\\circ) = 22(cos0\u2218+isin0\u2218)=2 (which is the principal real root).<\/li>\n\n\n\n
    • For k=3k = 3k=3, the root is 2(cos\u2061180\u2218+isin\u2061180\u2218)=\u221222(\\cos 180^\\circ + i \\sin 180^\\circ) = -22(cos180\u2218+isin180\u2218)=\u22122 (which is the other real root).<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

      Thus, the two real sixth roots of 64 are 222 and \u22122-2\u22122.<\/p>\n\n\n\n

      Conclusion:<\/h3>\n\n\n\n

      There are exactly 2 real sixth roots<\/strong> of 64: 222 and \u22122-2\u22122.<\/p>\n\n\n\n

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

      How many real sixth roots does 64 have? 0, or 1, or 2 The Correct Answer and Explanation is: The number 64 has 2 real sixth roots. Here’s an explanation: To find the sixth roots of a number, we want to solve for xxx in the equation x6=64x^6 = 64×6=64. We can express 64 as […]<\/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-266687","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266687","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=266687"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266687\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266687"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266687"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266687"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}