{"id":235232,"date":"2025-06-15T03:42:04","date_gmt":"2025-06-15T03:42:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=235232"},"modified":"2025-06-15T03:42:06","modified_gmt":"2025-06-15T03:42:06","slug":"find-the-fifth-root-of-32i-that-graphs-in-the-first-quadrant","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/15\/find-the-fifth-root-of-32i-that-graphs-in-the-first-quadrant\/","title":{"rendered":"Find the fifth root of 32i that graphs in the first quadrant"},"content":{"rendered":"\n

Find the fifth root of 32i that graphs in the first quadrant: 2 ](cos[ Jo + i sin[ 19) Use degree measure:<\/p>\n\n\n\n

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

Correct Answer:<\/strong><\/p>\n\n\n\n

The fifth root of 32i32i32i that lies in the first quadrant<\/strong> is: 2(cos\u2061(18\u2218)+isin\u2061(18\u2218))2 \\left( \\cos\\left(18^\\circ\\right) + i \\sin\\left(18^\\circ\\right) \\right)2(cos(18\u2218)+isin(18\u2218))<\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/strong><\/p>\n\n\n\n

To find the fifth roots of a complex number like 32i32i32i, the number must first be written in polar form<\/strong>. A complex number z=r(cos\u2061\u03b8+isin\u2061\u03b8)z = r(\\cos \\theta + i \\sin \\theta)z=r(cos\u03b8+isin\u03b8) can be rooted using De Moivre\u2019s Theorem: z1\/n=r1\/n(cos\u2061(\u03b8+360\u2218kn)+isin\u2061(\u03b8+360\u2218kn)),k=0,1,…,n\u22121z^{1\/n} = r^{1\/n} \\left( \\cos\\left(\\frac{\\theta + 360^\\circ k}{n} \\right) + i \\sin\\left(\\frac{\\theta + 360^\\circ k}{n} \\right) \\right), \\quad k = 0, 1, …, n-1z1\/n=r1\/n(cos(n\u03b8+360\u2218k\u200b)+isin(n\u03b8+360\u2218k\u200b)),k=0,1,…,n\u22121<\/p>\n\n\n\n

First, convert 32i32i32i to polar form. The modulus is: r=\u222332i\u2223=02+322=32r = |32i| = \\sqrt{0^2 + 32^2} = 32r=\u222332i\u2223=02+322\u200b=32<\/p>\n\n\n\n

The angle \u03b8\\theta\u03b8 (argument) is the angle from the positive real axis to the vector 32i32i32i, which is located on the positive imaginary axis. So: \u03b8=90\u2218\\theta = 90^\\circ\u03b8=90\u2218<\/p>\n\n\n\n

Now, apply De Moivre\u2019s Theorem to find the fifth roots: Let n=5,r1\/5=321\/5=2\\text{Let } n = 5,\\quad r^{1\/5} = 32^{1\/5} = 2Let n=5,r1\/5=321\/5=2<\/p>\n\n\n\n

Now compute the five roots using: zk=2(cos\u2061(90\u2218+360\u2218k5)+isin\u2061(90\u2218+360\u2218k5))z_k = 2 \\left( \\cos\\left( \\frac{90^\\circ + 360^\\circ k}{5} \\right) + i \\sin\\left( \\frac{90^\\circ + 360^\\circ k}{5} \\right) \\right)zk\u200b=2(cos(590\u2218+360\u2218k\u200b)+isin(590\u2218+360\u2218k\u200b))<\/p>\n\n\n\n

for k=0,1,2,3,4k = 0, 1, 2, 3, 4k=0,1,2,3,4. The angles are:<\/p>\n\n\n\n

    \n
  • k=0\u219290\u22185=18\u2218k = 0 \\rightarrow \\frac{90^\\circ}{5} = 18^\\circk=0\u2192590\u2218\u200b=18\u2218<\/li>\n\n\n\n
  • k=1\u2192450\u22185=90\u2218k = 1 \\rightarrow \\frac{450^\\circ}{5} = 90^\\circk=1\u21925450\u2218\u200b=90\u2218<\/li>\n\n\n\n
  • k=2\u2192810\u22185=162\u2218k = 2 \\rightarrow \\frac{810^\\circ}{5} = 162^\\circk=2\u21925810\u2218\u200b=162\u2218<\/li>\n\n\n\n
  • k=3\u21921170\u22185=234\u2218k = 3 \\rightarrow \\frac{1170^\\circ}{5} = 234^\\circk=3\u219251170\u2218\u200b=234\u2218<\/li>\n\n\n\n
  • k=4\u21921530\u22185=306\u2218k = 4 \\rightarrow \\frac{1530^\\circ}{5} = 306^\\circk=4\u219251530\u2218\u200b=306\u2218<\/li>\n<\/ul>\n\n\n\n

    Only the root at 18\u00b0<\/strong> lies in the first quadrant<\/strong> (where both cosine and sine are positive). Therefore, the correct fifth root is: 2(cos\u2061(18\u2218)+isin\u2061(18\u2218))\\boxed{2 \\left( \\cos(18^\\circ) + i \\sin(18^\\circ) \\right)}2(cos(18\u2218)+isin(18\u2218))\u200b<\/p>\n\n\n\n

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

    Find the fifth root of 32i that graphs in the first quadrant: 2 ](cos[ Jo + i sin[ 19) Use degree measure: The Correct Answer and Explanation is: Correct Answer: The fifth root of 32i32i32i that lies in the first quadrant is: 2(cos\u2061(18\u2218)+isin\u2061(18\u2218))2 \\left( \\cos\\left(18^\\circ\\right) + i \\sin\\left(18^\\circ\\right) \\right)2(cos(18\u2218)+isin(18\u2218)) Explanation: To find the fifth roots […]<\/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-235232","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235232","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=235232"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235232\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=235232"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=235232"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=235232"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}