{"id":266683,"date":"2025-07-23T06:29:06","date_gmt":"2025-07-23T06:29:06","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=266683"},"modified":"2025-07-23T06:29:08","modified_gmt":"2025-07-23T06:29:08","slug":"identify-all-real-sixth-roots-of-262144-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/identify-all-real-sixth-roots-of-262144-2\/","title":{"rendered":"Identify all real sixth roots of 262,144 2."},"content":{"rendered":"\n

Identify all real sixth roots of 262,144 2.) Identify all real square roots of -64<\/p>\n\n\n\n

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

1. Sixth Roots of 262,144:<\/h3>\n\n\n\n

To find the sixth roots of 262,144, we first express 262,144 as a power of 2:262,144=218262,144 = 2^{18}262,144=218<\/p>\n\n\n\n

The sixth roots are the values of xxx that satisfy:x6=262,144=218x^6 = 262,144 = 2^{18}x6=262,144=218<\/p>\n\n\n\n

This implies that x=218\/6=23=8x = 2^{18\/6} = 2^3 = 8x=218\/6=23=8. Thus, the principal sixth root is 888. However, there are six possible real roots for a sixth root because the equation x6=218x^6 = 2^{18}x6=218 can have multiple solutions due to the periodic nature of complex roots. Specifically, the sixth roots of a positive number rrr are given by:xk=r1\/6\u22c5ei(2k\u03c0\/6)x_k = r^{1\/6} \\cdot e^{i(2k\\pi\/6)}xk\u200b=r1\/6\u22c5ei(2k\u03c0\/6)<\/p>\n\n\n\n

For real roots, the possible values are 888 (since we’re dealing with a positive number and the principal root is real) and the negative of the principal root, \u22128-8\u22128.<\/p>\n\n\n\n

Thus, the real sixth roots of 262,144 are:8 and \u22128.8 \\text{ and } -8.8 and \u22128.<\/p>\n\n\n\n

2. Square Roots of -64:<\/h3>\n\n\n\n

For square roots of a negative number, we use the imaginary unit iii, where i=\u22121i = \\sqrt{-1}i=\u22121\u200b.<\/p>\n\n\n\n

We can write \u221264-64\u221264 as:\u221264=64\u00d7(\u22121)-64 = 64 \\times (-1)\u221264=64\u00d7(\u22121)<\/p>\n\n\n\n

The square roots of 646464 are 888 and \u22128-8\u22128. So, the square roots of \u221264-64\u221264 are:\u00b18i\\pm 8i\u00b18i<\/p>\n\n\n\n

Thus, the real square roots of \u221264-64\u221264 do not exist, but the complex square roots are:8i and \u22128i.8i \\text{ and } -8i.8i and \u22128i.<\/p>\n\n\n\n

Summary:<\/h3>\n\n\n\n
    \n
  • The sixth roots of 262,144<\/strong> are 888 and \u22128-8\u22128.<\/li>\n\n\n\n
  • The square roots of \u221264-64\u221264<\/strong> are 8i8i8i and \u22128i-8i\u22128i, which are purely imaginary.<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Identify all real sixth roots of 262,144 2.) Identify all real square roots of -64 The Correct Answer and Explanation is: 1. Sixth Roots of 262,144: To find the sixth roots of 262,144, we first express 262,144 as a power of 2:262,144=218262,144 = 2^{18}262,144=218 The sixth roots are the values of xxx that satisfy:x6=262,144=218x^6 = […]<\/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-266683","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266683","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=266683"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266683\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266683"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266683"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}