{"id":235227,"date":"2025-06-15T03:37:49","date_gmt":"2025-06-15T03:37:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=235227"},"modified":"2025-06-15T03:37:52","modified_gmt":"2025-06-15T03:37:52","slug":"find-the-indicated-roots-and-graph-them-in-the-complex-plane","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/15\/find-the-indicated-roots-and-graph-them-in-the-complex-plane\/","title":{"rendered":"Find the indicated roots and graph them in the complex plane."},"content":{"rendered":"\n<p>Find the indicated roots and graph them in the complex plane. The fifth roots of 32 Im W1 i Wo W2 -2 -1 W2 -2 W3 W1 W3 Im WA Re 2 Wo i -2 W2 Im W1 i W3 Im W1 W2 W2 Wo Re 2 W4 Wo Re Re -2 2 2 -1 W4 W3 W4 Referring to the labels from the graph you selected above, give the exact roots. (Express<br>in radians.)<\/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\/image-518.png\" alt=\"\" class=\"wp-image-235228\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the fifth roots of 323232, express the number in polar (trigonometric) form. The number 32 is a real number with no imaginary part:32=32(cos\u2061(0)+isin\u2061(0))32 = 32(\\cos(0) + i\\sin(0))32=32(cos(0)+isin(0))<\/p>\n\n\n\n<p>Let the complex number be z=r(cos\u2061(\u03b8)+isin\u2061(\u03b8))z = r(\\cos(\\theta) + i\\sin(\\theta))z=r(cos(\u03b8)+isin(\u03b8)), where r=32r = 32r=32 and \u03b8=0\\theta = 0\u03b8=0.<\/p>\n\n\n\n<p>To find the fifth roots of a complex number, apply De Moivre&#8217;s Theorem:wk=325(cos\u2061(2\u03c0k5)+isin\u2061(2\u03c0k5)),k=0,1,2,3,4w_k = \\sqrt[5]{32} \\left( \\cos\\left(\\frac{2\\pi k}{5}\\right) + i\\sin\\left(\\frac{2\\pi k}{5}\\right) \\right), \\quad k = 0, 1, 2, 3, 4wk\u200b=532\u200b(cos(52\u03c0k\u200b)+isin(52\u03c0k\u200b)),k=0,1,2,3,4<\/p>\n\n\n\n<p>Since 325=2\\sqrt[5]{32} = 2532\u200b=2, the roots are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>w0=2(cos\u2061(2\u03c0(0)5)+isin\u2061(2\u03c0(0)5))=2(cos\u2061(0)+isin\u2061(0))=2w_0 = 2\\left( \\cos\\left(\\frac{2\\pi(0)}{5} \\right) + i\\sin\\left( \\frac{2\\pi(0)}{5} \\right) \\right) = 2(\\cos(0) + i\\sin(0)) = 2w0\u200b=2(cos(52\u03c0(0)\u200b)+isin(52\u03c0(0)\u200b))=2(cos(0)+isin(0))=2<\/li>\n\n\n\n<li>w1=2(cos\u2061(2\u03c05)+isin\u2061(2\u03c05))w_1 = 2\\left( \\cos\\left(\\frac{2\\pi}{5} \\right) + i\\sin\\left( \\frac{2\\pi}{5} \\right) \\right)w1\u200b=2(cos(52\u03c0\u200b)+isin(52\u03c0\u200b))<\/li>\n\n\n\n<li>w2=2(cos\u2061(4\u03c05)+isin\u2061(4\u03c05))w_2 = 2\\left( \\cos\\left(\\frac{4\\pi}{5} \\right) + i\\sin\\left( \\frac{4\\pi}{5} \\right) \\right)w2\u200b=2(cos(54\u03c0\u200b)+isin(54\u03c0\u200b))<\/li>\n\n\n\n<li>w3=2(cos\u2061(6\u03c05)+isin\u2061(6\u03c05))w_3 = 2\\left( \\cos\\left(\\frac{6\\pi}{5} \\right) + i\\sin\\left( \\frac{6\\pi}{5} \\right) \\right)w3\u200b=2(cos(56\u03c0\u200b)+isin(56\u03c0\u200b))<\/li>\n\n\n\n<li>w4=2(cos\u2061(8\u03c05)+isin\u2061(8\u03c05))w_4 = 2\\left( \\cos\\left(\\frac{8\\pi}{5} \\right) + i\\sin\\left( \\frac{8\\pi}{5} \\right) \\right)w4\u200b=2(cos(58\u03c0\u200b)+isin(58\u03c0\u200b))<\/li>\n<\/ul>\n\n\n\n<p>From the graph selected (bottom right), the roots appear labeled correctly in order around the unit circle:<\/p>\n\n\n\n<p><strong>Exact roots:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>w0=2(cos\u2061(0\u03c05)+isin\u2061(0\u03c05))w_0 = 2\\left( \\cos\\left(\\frac{0\\pi}{5}\\right) + i\\sin\\left(\\frac{0\\pi}{5}\\right) \\right)w0\u200b=2(cos(50\u03c0\u200b)+isin(50\u03c0\u200b))<\/li>\n\n\n\n<li>w1=2(cos\u2061(2\u03c05)+isin\u2061(2\u03c05))w_1 = 2\\left( \\cos\\left(\\frac{2\\pi}{5}\\right) + i\\sin\\left(\\frac{2\\pi}{5}\\right) \\right)w1\u200b=2(cos(52\u03c0\u200b)+isin(52\u03c0\u200b))<\/li>\n\n\n\n<li>w2=2(cos\u2061(4\u03c05)+isin\u2061(4\u03c05))w_2 = 2\\left( \\cos\\left(\\frac{4\\pi}{5}\\right) + i\\sin\\left(\\frac{4\\pi}{5}\\right) \\right)w2\u200b=2(cos(54\u03c0\u200b)+isin(54\u03c0\u200b))<\/li>\n\n\n\n<li>w3=2(cos\u2061(6\u03c05)+isin\u2061(6\u03c05))w_3 = 2\\left( \\cos\\left(\\frac{6\\pi}{5}\\right) + i\\sin\\left(\\frac{6\\pi}{5}\\right) \\right)w3\u200b=2(cos(56\u03c0\u200b)+isin(56\u03c0\u200b))<\/li>\n\n\n\n<li>w4=2(cos\u2061(8\u03c05)+isin\u2061(8\u03c05))w_4 = 2\\left( \\cos\\left(\\frac{8\\pi}{5}\\right) + i\\sin\\left(\\frac{8\\pi}{5}\\right) \\right)w4\u200b=2(cos(58\u03c0\u200b)+isin(58\u03c0\u200b))<\/li>\n<\/ul>\n\n\n\n<p>These roots are evenly spaced around the origin in the complex plane, forming a regular pentagon. Each root lies at an angle increment of 2\u03c05\\frac{2\\pi}{5}52\u03c0\u200b radians from the previous one, creating symmetry.<\/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-banner5-564.jpeg\" alt=\"\" class=\"wp-image-235229\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the indicated roots and graph them in the complex plane. The fifth roots of 32 Im W1 i Wo W2 -2 -1 W2 -2 W3 W1 W3 Im WA Re 2 Wo i -2 W2 Im W1 i W3 Im W1 W2 W2 Wo Re 2 W4 Wo Re Re -2 2 2 -1 [&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-235227","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235227","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=235227"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235227\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=235227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=235227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=235227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}