{"id":266678,"date":"2025-07-23T06:27:17","date_gmt":"2025-07-23T06:27:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=266678"},"modified":"2025-07-23T06:27:19","modified_gmt":"2025-07-23T06:27:19","slug":"find-all-the-complex-roots","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/find-all-the-complex-roots\/","title":{"rendered":"Find all the complex roots."},"content":{"rendered":"\n

Find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64<\/p>\n\n\n\n

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

To find the complex sixth roots of 646464, we can represent 646464 in polar form and use De Moivre’s Theorem.<\/p>\n\n\n\n

Step 1: Express 64 in Polar Form<\/h3>\n\n\n\n

First, write 646464 in polar form. 646464 is a real number, so its polar form is:64=64\u22c5(cos\u20610\u2218+isin\u20610\u2218)64 = 64 \\cdot (\\cos 0^\\circ + i \\sin 0^\\circ)64=64\u22c5(cos0\u2218+isin0\u2218)<\/p>\n\n\n\n

Here, the modulus r=64r = 64r=64, and the argument \u03b8=0\u2218\\theta = 0^\\circ\u03b8=0\u2218.<\/p>\n\n\n\n

Step 2: Apply De Moivre\u2019s Theorem<\/h3>\n\n\n\n

De Moivre\u2019s Theorem states that the nnn-th roots of a complex number r(cos\u2061\u03b8+isin\u2061\u03b8)r(\\cos \\theta + i \\sin \\theta)r(cos\u03b8+isin\u03b8) are given by:Roots=rn[cos\u2061(\u03b8+360kn)+isin\u2061(\u03b8+360kn)]\\text{Roots} = \\sqrt[n]{r} \\left[ \\cos\\left(\\frac{\\theta + 360k}{n}\\right) + i \\sin\\left(\\frac{\\theta + 360k}{n}\\right) \\right]Roots=nr\u200b[cos(n\u03b8+360k\u200b)+isin(n\u03b8+360k\u200b)]<\/p>\n\n\n\n

where k=0,1,2,\u2026,n\u22121k = 0, 1, 2, \\dots, n-1k=0,1,2,\u2026,n\u22121.<\/p>\n\n\n\n

In our case, r=64r = 64r=64 and n=6n = 6n=6, so the sixth roots of 646464 are:646[cos\u2061(0+360k6)+isin\u2061(0+360k6)]\\sqrt[6]{64} \\left[ \\cos\\left(\\frac{0 + 360k}{6}\\right) + i \\sin\\left(\\frac{0 + 360k}{6}\\right) \\right]664\u200b[cos(60+360k\u200b)+isin(60+360k\u200b)]<\/p>\n\n\n\n

Since 646=2\\sqrt[6]{64} = 2664\u200b=2, the formula becomes:2[cos\u2061(360k6)+isin\u2061(360k6)]2 \\left[ \\cos\\left(\\frac{360k}{6}\\right) + i \\sin\\left(\\frac{360k}{6}\\right) \\right]2[cos(6360k\u200b)+isin(6360k\u200b)]<\/p>\n\n\n\n

Step 3: Calculate the Roots<\/h3>\n\n\n\n

Now, substitute different values of kkk (from 0 to 5) into the equation.<\/p>\n\n\n\n

    \n
  • For k=0k = 0k=0: 2[cos\u2061(0\u2218)+isin\u2061(0\u2218)]=2\u00d7(1+0i)=22 \\left[ \\cos(0^\\circ) + i \\sin(0^\\circ) \\right] = 2 \\times (1 + 0i) = 22[cos(0\u2218)+isin(0\u2218)]=2\u00d7(1+0i)=2<\/li>\n\n\n\n
  • For k=1k = 1k=1: 2[cos\u2061(60\u2218)+isin\u2061(60\u2218)]=2[12+i32]=1+i32 \\left[ \\cos(60^\\circ) + i \\sin(60^\\circ) \\right] = 2 \\left[ \\frac{1}{2} + i \\frac{\\sqrt{3}}{2} \\right] = 1 + i\\sqrt{3}2[cos(60\u2218)+isin(60\u2218)]=2[21\u200b+i23\u200b\u200b]=1+i3\u200b<\/li>\n\n\n\n
  • For k=2k = 2k=2: 2[cos\u2061(120\u2218)+isin\u2061(120\u2218)]=2[\u221212+i32]=\u22121+i32 \\left[ \\cos(120^\\circ) + i \\sin(120^\\circ) \\right] = 2 \\left[ -\\frac{1}{2} + i \\frac{\\sqrt{3}}{2} \\right] = -1 + i\\sqrt{3}2[cos(120\u2218)+isin(120\u2218)]=2[\u221221\u200b+i23\u200b\u200b]=\u22121+i3\u200b<\/li>\n\n\n\n
  • For k=3k = 3k=3: 2[cos\u2061(180\u2218)+isin\u2061(180\u2218)]=2\u00d7(\u22121+0i)=\u221222 \\left[ \\cos(180^\\circ) + i \\sin(180^\\circ) \\right] = 2 \\times (-1 + 0i) = -22[cos(180\u2218)+isin(180\u2218)]=2\u00d7(\u22121+0i)=\u22122<\/li>\n\n\n\n
  • For k=4k = 4k=4: 2[cos\u2061(240\u2218)+isin\u2061(240\u2218)]=2[\u221212\u2212i32]=\u22121\u2212i32 \\left[ \\cos(240^\\circ) + i \\sin(240^\\circ) \\right] = 2 \\left[ -\\frac{1}{2} – i \\frac{\\sqrt{3}}{2} \\right] = -1 – i\\sqrt{3}2[cos(240\u2218)+isin(240\u2218)]=2[\u221221\u200b\u2212i23\u200b\u200b]=\u22121\u2212i3\u200b<\/li>\n\n\n\n
  • For k=5k = 5k=5: 2[cos\u2061(300\u2218)+isin\u2061(300\u2218)]=2[12\u2212i32]=1\u2212i32 \\left[ \\cos(300^\\circ) + i \\sin(300^\\circ) \\right] = 2 \\left[ \\frac{1}{2} – i \\frac{\\sqrt{3}}{2} \\right] = 1 – i\\sqrt{3}2[cos(300\u2218)+isin(300\u2218)]=2[21\u200b\u2212i23\u200b\u200b]=1\u2212i3\u200b<\/li>\n<\/ul>\n\n\n\n

    Step 4: List the Roots<\/h3>\n\n\n\n

    The six complex roots of 646464 are:<\/p>\n\n\n\n

      \n
    1. 222<\/li>\n\n\n\n
    2. 1+i31 + i\\sqrt{3}1+i3\u200b<\/li>\n\n\n\n
    3. \u22121+i3-1 + i\\sqrt{3}\u22121+i3\u200b<\/li>\n\n\n\n
    4. \u22122-2\u22122<\/li>\n\n\n\n
    5. \u22121\u2212i3-1 – i\\sqrt{3}\u22121\u2212i3\u200b<\/li>\n\n\n\n
    6. 1\u2212i31 – i\\sqrt{3}1\u2212i3\u200b<\/li>\n<\/ol>\n\n\n\n

      These are the complex sixth roots of 646464 in rectangular form.<\/p>\n\n\n\n

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

      The six complex roots of 646464 are:2,1+i3,\u22121+i3,\u22122,\u22121\u2212i3,1\u2212i32, \\quad 1 + i\\sqrt{3}, \\quad -1 + i\\sqrt{3}, \\quad -2, \\quad -1 – i\\sqrt{3}, \\quad 1 – i\\sqrt{3}2,1+i3\u200b,\u22121+i3\u200b,\u22122,\u22121\u2212i3\u200b,1\u2212i3\u200b<\/p>\n\n\n\n

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

      Find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64 The Correct Answer and Explanation is: To find the complex sixth roots of 646464, we can represent 646464 in polar form and use De Moivre’s Theorem. Step 1: Express 64 in Polar […]<\/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-266678","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266678","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=266678"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266678\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266678"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266678"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266678"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}