Lesson Solved problems on taking roots of complex numbers
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<H2>Solved problems on taking roots of complex numbers</H2> In this lesson you will find solved typical basic problems on taking roots of complex numbers. The solutions are based on a general theory and the formulas of the lesson <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-root-of-a-complex-number.lesson>How to take a root of a complex number</A> in this site. <H3>Problem 1</H3>What are the three cube roots of "i"? Express the roots in rectangular form. <B>Solution</B> <pre> The trigonometric form of the complex number "i" is cos(90°) + i*sin(90°). The modulus of "i" is 1, the argument is 90° = {{{pi/2}}}. According to the general theory, there are three complex cube roots of "i". They have the modulus of {{{root(3,1)}}} = 1. The first cube root has the argument of 30° = {{{pi/6}}}, one third of the argument of "i". Each next cube root has the argument in {{{360^o/3}}} = 120° = {{{2pi/3}}} more than the previous one. Thus the tree complex roots are 1) {{{cos((1/3)*(pi/2)) + i*sin((1/3)*(pi/2))}}} = {{{cos(pi/6) + i*sin(pi/6)}}} = cos(30°) + i*sin(30°) = {{{sqrt(3)/2 + (1/2)*i}}}; 2) {{{cos(pi/6 + 2pi/3) + i*sin(pi/6 + 2pi/3)}}} = cos(30°+120°) + i*sin(30° + 120°) = cos(150°) + i*sin(150°) = {{{-sqrt(3)/2 + (1/2)*i}}}; 3) {{{cos(pi/6 + 2*(2pi/3)) + i*sin(pi/6 + 2*(2pi/3))}}} = cos(30°+240°) + i*sin(30° + 240°) = cos(270°) + i*sin(270°) = {{{0 - 1*i}}} = {{{-i}}}. </pre> <H3>Problem 2</H3>What are the cube roots of {{{(1/2-(sqrt(3)/2) *i)}}} in complex domain? <B>Solution</B> <pre> The trigonometric form of the number {{{1/2-(sqrt(3)/2)*i}}} is cos(300°) + i*sin(300°). The modulus of the number is 1, the argument is 300° = {{{5pi/3}}}. According to the general theory, there are three complex cube roots of "i". They have the modulus of {{{root(3,1)}}} = 1. The first cube root has the argument of 100° = {{{5pi/9}}}, one third of the argument of the original number. Each next cube root has the argument in {{{360^o/3}}} = 120° = {{{2pi/3}}} more than the previous one. Thus the tree complex roots are 1) cos(100°) + i*sin(100°) 2) cos(100°+120°) + i*sin(100°+120°) = cos(220°) + i*sin(220°) 3) cos(100° + 240°) + i*sin(100°+240°) = cos(340°) + i*sin(340°) </pre> <H3>Problem 3</H3>Determine the fourth roots of -16 in complex domain. <B>Solution</B> <pre> In the complex plane, -16 = {{{2^4*(cos(pi) + i*sin(pi))}}}. The modulus of -16 is {{{2^4}}}, the argument is 180° = {{{pi}}}. According to the general theory, there are four complex fourth roots of the number -16. They have the modulus of {{{root(4,16)}}} = 2. The first fourth root has the argument of 45° = {{{pi/4}}}, one fourth of the argument of -16. Each next fourth root has the argument in {{{360^o/4}}} = 90° = {{{pi/2}}} more than the previous one. Thus the four complex roots are 1) {{{2*(cos(pi/4) + i*sin(pi/4))}}} = {{{2*(sqrt(2)/2+i*sqrt(2)/2)}}} = {{{sqrt(2)+i*sqrt(2)}}}; 2) {{{2*(cos(pi/4 + 2pi/4) + i*sin(pi/4+2pi/4))}}} = {{{2*(cos(3pi/4)+i*sin(3pi/4))}}} = {{{2*(-sqrt(2)/2 + i*(sqrt(2)/2))}}} = {{{-sqrt(2) + i*sqrt(2)}}}; 3) {{{2*(cos(pi/4 + 4pi/4) + i*sin(pi/4+4pi/4))}}} = {{{2*(cos(5pi/4)+i*sin(5pi/4))}}} = {{{2*(-sqrt(2)/2 + i*(-sqrt(2)/2)))}}} = {{{-sqrt(2) - i*sqrt(2)}}}; 4) {{{2*(cos(pi/4 + 6pi/4) + i*sin(pi/4+6pi/4))}}} = {{{2*(cos(7pi/4)+i*sin(7pi/4))}}} = {{{2*(sqrt(2)/2 + i*(-sqrt(2)/2)))}}} = {{{sqrt(2) - i*sqrt(2)}}}. <U>Answer</U>. The four values of fourth root of -16 are {{{sqrt(2)+i*sqrt(2)}}}, {{{-sqrt(2) + i*sqrt(2)}}}, {{{-sqrt(2) - i*sqrt(2)}}} and {{{sqrt(2) - i*sqrt(2)}}}. </pre> <H3>Problem 4</H3>Find complex roots of degree 4 of -256 in complex domain. <B>Solution</B> <pre> In the complex plane, -256 = 256*(cos(180°) + i*sin(180°)) = {{{4^4*(cos(pi) + i*sin(pi))}}}. The modulus of -256 is {{{4^4}}}, the argument is 180° = {{{pi}}}. According to the general theory, there are four complex fourth roots of the number -256. They have the modulus of {{{root(4,256)}}} = 4. The first fourth root has the argument of 45° = {{{pi/4}}}, one fourth of the argument of -256. Each next fourth root has the argument in {{{360^o/4}}} = 90° = {{{pi/2}}} more than the previous one. Thus the four complex roots are 1) 4(cos(45°) + i*sin(45°)) = {{{4(sqrt(2)/2 + i*(sqrt(2)/2))}}} = {{{2sqrt(2) + 2sqrt(2)*i}}}; 2) 4(cos(45°+90°) + i*sin(45°+90°)) = 4(cos(135°) + i*sin(135°)) = {{{-4*sqrt(2)/2 + ((4*sqrt(2))/2)*i}}} = {{{-2sqrt(2) + 2sqrt(2)*i}}}; 3) 4(cos(45°+180°) + i*sin(45°+180°)) = 4(cos(225°) + i*sin(225°)) = {{{-4*sqrt(2)/2 - (4*sqrt(2)/2)*i}}} = {{{-2sqrt(2) - 2sqrt(2)*i}}}; 4) 4(cos(45°+270°) + i*sin(45°+270°)) = 4(cos(315°) + i*sin(315°)) = {{{4*sqrt(2)/2 - (4*sqrt(2)/2)*i}}} = {{{2sqrt(2) - 2sqrt(2)*i}}}. </pre> <H3>Problem 5</H3>What are the 5th roots of 1 in complex domain? <B>Solution</B> <pre> {{{root(5,1)}}} has 5 values in complex domain. 1) 1 (= cos(0°) + i*sin(0°) ), 2) cos(72°) + i*sin(72°), 3) cos(144°) + i*sin(144°), 4) cos(216°) + i*sin(216°), 5) cos(288°) + i*sin(288°). </pre> <H3>Problem 6</H3>What are the fifth roots of -32i? <B>Solution</B> <pre> Notice that -32i = {{{2^5*cis(270^o)}}}. Therefore, due to DeMoivre formulas, the 5-th degree roots of -32i are (1) 2cis(54°); ( notice that 54° = {{{270^o/5}}} ); (2) {{{2cis(54^o + 360^o/5)}}} = = 2cis(54° + 72°) = 2cis(126°); ( Notice that 72° = {{{360^o/5}}} ) (3) {{{2cis(54^o + 2*(360^o/5))}}} = 2cis(54° + 2*72°) = 2cis(198°); (4) {{{2cis(54^o + 3*(360^o/5))}}} = 2cis(54° + 3*72°) = 2cis(270°); (5) {{{2cis(54^o + 4*(360^o/5))}}} = 2cis(54° + 4*72°) = 2cis(342°). </pre> <H3>Problem 7</H3>Find the sixth roots of 64 in complex domain. <B>Solution</B> <pre> {{{root(6,64)}}} has 6 values in complex domain. 1) 2*(cos(0°) + i*sin(0°)) = 2, 2) 2*(cos(60) + i*sin(60°)) = {{{2*((1/2) + i*(sqrt(3)/2))}}} = {{{1 + i*sqrt(3))}}}, 3) 2*(cos(120°) + i*sin(120°)) = {{{2*((-1/2) + i*(sqrt(3)/2))}}} = {{{-1 + i*sqrt(3))}}}, 4) 2*(cos(180°) + i*sin(180°)) = -2, 5) 2*(cos(240°) + i*sin(240°)) = {{{2*((-1/2) - i*(sqrt(3)/2))}}} = {{{-1 - i*sqrt(3))}}}, 6) 2*(cos(300°) + i*sin(300°)) = {{{2*((1/2) - i*(sqrt(3)/2))}}} = {{{1 - i*sqrt(3))}}}. </pre> My lessons on complex numbers in this site are - <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetic operations on them</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson> Complex plane</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-in-complex-plane-.lesson>Multiplication and division of complex numbers in complex plane</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Raising-a-complex-number-to-an-integer-power.lesson>Raising a complex number to an integer power</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-root-of-a-complex-number.lesson>How to take a root of a complex number</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-real-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with real coefficients on complex domain</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-square-root-of-a-complex-number.lesson>How to take a square root of a complex number</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-complex-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with complex coefficients on complex domain</A> - Solved problems on taking roots of complex numbers (this lesson) - <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-arithmetic-operations-on-complex-numbers.lesson>Solved problems on arithmetic operations on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problem-on-taking-square-roots-of-complex-numbers.lesson>Solved problem on taking square root of complex number</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Solving-polynomial--equations-in-complex-domain.lesson>Solving polynomial equations in complex domain</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Miscellaneous-problems-on-complex-numbers.lesson>Miscellaneous problems on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Advanced-problem-in-complex-numbers.lesson>Advanced problems on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-de%27Moivre-formula.lesson>Solved problems on de'Moivre formula</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Proving-identities-using-complex-numbers.lesson>Proving identities using complex numbers</A> - <A HREF=https://www.algebra.com/tutors/18-Calculating-1sin%281%B0%29%2B2sin%282%B0%29%2B3sin%283%B0%29%2B-%2B180sin%28180%B0%29.lesson>Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/An-equation-in-complex-numbers-which-HAS-NO-a-solution.lesson>A curious example of an equation in complex numbers which HAS NO a solution</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Solving-one-non-standard-equation-in-complex-numbers.lesson>Solving non-standard equations in complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Upper-level-problem-on-complex-numbers.lesson>Upper level problem on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Determine-locus-of-points-using-complex-numbers.lesson>Determine locus of points using complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Joke-problems-on-complex-numbers.lesson>Joke problems on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Review-of-lessons-on-complex-numbers.lesson>OVERVIEW of lessons on complex numbers</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
