Question 1143637
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<pre>
I will use trigonometric form of complex numbers.


Let  x + iy = r*(cos(a) + i*sin(a)),

where r > 0 is the modulus and "a" is a polar angle.


Then x - iy = r*(cos(a) - i*sin(a)).


According to de Moivre's theorem,

    {{{(x+iy)^5}}} = {{{r^5*(cos(5a) + i*sin(5a))}}},

    {{{(x-iy)^5}}} = {{{r^5*(cos(5a) - i*sin(5a))}}}.


Then

    {{{(x+iy)^5}}} + {{{(x-iy)^5}}} = {{{r^5*(cos(5a) + i*sin(5a))}}} + {{{r^5*(cos(5a) - i*sin(5a))}}} = {{{2r^5*cos(5a)}}}.


The problem asks me to find ONLY ONE ROOT  to equation  

    (x+iy)^5 + (x-iy)^5 = -8.     (1)


It is equivalent to find one root of the equation

    {{{2r^5*cos(5a)}}} = -8.               (2)


I will take  a = {{{pi/5}}} radians = {{{180/5}}} degrees = 36 degrees.


Then  cos(5a) = {{{cos(pi)}}} = -1,  and the equation (2)  takes the form


    {{{2r^5*(-1)}}} = -8,      

or, equivalently,

    {{{2r^5}}} = 8,            (3)

    {{{r^5}}} = 4,             (4)

    r = {{{root(5,4)}}}.            (5)


Thus  

    x + iy = {{{root(5,4)*(cos(pi/5) + i*sin(pi/5))}}} = {{{root(5,4)*(cos(36^o) + i*sin(36^o))}}}


is one possible root of the equation (1).
</pre>

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On complex numbers, &nbsp;there is a bunch of my lessons in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-taking-roots-of-complex-numbers.lesson>Solved problems on taking roots of complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Miscellaneous-problems-on-complex-numbers.lesson>Miscellaneous problems on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Advanced-problem-in-complex-numbers.lesson>Advanced problem on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-de%27Moivre-formula.lesson>Solved problems on de'Moivre formula</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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> 

&nbsp;&nbsp;&nbsp;&nbsp;- <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>



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Complex numbers</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.