Find the fifth roots of 243.<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n The problem is to find the fifth roots of 243. This means finding all complex numbers zz such that: z5=243z^5 = 243<\/p>\n\n\n\n 243 is a positive real number. It can be written as a power of 3: 243=35243 = 3^5<\/p>\n\n\n\n In polar (or complex exponential) form, a positive real number rr can be written as: r=r\u22c5(cos\u20610+isin\u20610)=rei\u22c50r = r \\cdot (\\cos 0 + i \\sin 0) = r e^{i \\cdot 0}<\/p>\n\n\n\n So, 243=35=243ei\u22c50243 = 3^5 = 243 e^{i \\cdot 0}<\/p>\n\n\n\n The general formula for the nnth roots of a complex number rei\u03b8r e^{i \\theta} is: zk=r1\/nei(\u03b8+2\u03c0kn)fork=0,1,2,…,n\u22121z_k = r^{1\/n} e^{i \\left(\\frac{\\theta + 2\\pi k}{n}\\right)} \\quad \\text{for} \\quad k = 0, 1, 2, …, n-1<\/p>\n\n\n\n Here, n=5n = 5, r=243r = 243, and \u03b8=0\\theta = 0.<\/p>\n\n\n\n Calculate the magnitude of the roots: r1\/5=2431\/5=(35)1\/5=3r^{1\/5} = 243^{1\/5} = (3^5)^{1\/5} = 3<\/p>\n\n\n\n Using the formula: zk=3ei0+2\u03c0k5=3(cos\u20612\u03c0k5+isin\u20612\u03c0k5)z_k = 3 e^{i \\frac{0 + 2\\pi k}{5}} = 3 \\left(\\cos \\frac{2\\pi k}{5} + i \\sin \\frac{2\\pi k}{5}\\right)<\/p>\n\n\n\n for k=0,1,2,3,4k = 0, 1, 2, 3, 4.<\/p>\n\n\n\n The roots are:<\/p>\n\n\n\n z0=3(cos\u20610+isin\u20610)=3(1+0i)=3z_0 = 3(\\cos 0 + i \\sin 0) = 3(1 + 0i) = 3<\/p>\n\n\n\n z1=3(cos\u20612\u03c05+isin\u20612\u03c05)z_1 = 3 \\left(\\cos \\frac{2\\pi}{5} + i \\sin \\frac{2\\pi}{5}\\right)<\/p>\n\n\n\n z2=3(cos\u20614\u03c05+isin\u20614\u03c05)z_2 = 3 \\left(\\cos \\frac{4\\pi}{5} + i \\sin \\frac{4\\pi}{5}\\right)<\/p>\n\n\n\n z3=3(cos\u20616\u03c05+isin\u20616\u03c05)z_3 = 3 \\left(\\cos \\frac{6\\pi}{5} + i \\sin \\frac{6\\pi}{5}\\right)<\/p>\n\n\n\n z4=3(cos\u20618\u03c05+isin\u20618\u03c05)z_4 = 3 \\left(\\cos \\frac{8\\pi}{5} + i \\sin \\frac{8\\pi}{5}\\right)<\/p>\n\n\n\n Using approximate values of cosine and sine:<\/p>\n\n\n\n So:<\/p>\n\n\n\n The number 243 is a perfect fifth power because it equals 353^5. Taking the fifth root gives a magnitude of 3. Because complex numbers have multiple roots equally spaced around the unit circle, there are five distinct fifth roots spaced evenly in angle by 2\u03c05\\frac{2\\pi}{5} radians. This is why the roots have the same magnitude but differ in their argument (angle). The principal root is the real positive root 33, while the other four are complex conjugates and symmetrically placed in the complex plane. This method uses De Moivre’s theorem, which connects the algebraic roots of complex numbers to trigonometric forms.<\/p>\n\n\n\n This explains why every complex number has exactly nn distinct nnth roots and shows the structure of these roots geometrically on the complex plane.<\/p>\n","protected":false},"excerpt":{"rendered":" Find the fifth roots of 243. The correct answer and explanation is: The problem is to find the fifth roots of 243. This means finding all complex numbers zz such that: z5=243z^5 = 243 Step 1: Express 243 in exponential form 243 is a positive real number. It can be written as a power of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-31193","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/31193","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=31193"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/31193\/revisions"}],"predecessor-version":[{"id":31195,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/31193\/revisions\/31195"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=31193"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=31193"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=31193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Express 243 in exponential form<\/h3>\n\n\n\n
Step 2: Use De Moivre’s theorem for roots<\/h3>\n\n\n\n
Step 3: Calculate the fifth roots<\/h3>\n\n\n\n
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Step 4: Approximate values<\/h3>\n\n\n\n
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Explanation<\/h3>\n\n\n\n