Question 1078164
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{{{highlight(cross(Do))}}} Does the sum of the n-th roots of a complex number z always {{{highlight(cross(equals))}}} equal zero? Show this using the examples below.

1. Find the cube roots of -i

2. Find the fourth roots of {{{ -1/2- (sqrt (3) /2)i }}}
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0.  Yes, this statement is TRUE.


<pre>
    The simplest way to explain/to illustrate it is to use square roots of complex number.

    For any complex number z, if {{{alpha}}} = {{{sqrt(z)}}} is a square root of z, then {{{-alpha}}} is also square root of z  (the second value),   

    and {{{alpha + (-alpha)}}} = 0.
</pre>


1. Find the cube roots of -i


<pre>
    -i = {{{cos(3pi/4) + i*sin(3pi/4)}}} = {{{cis(3pi/4)}}}.  

    Therefore, the three values of the cube root of -i are

        a)  {{{cis(pi/4)}}} = {{{cos(pi/4) + i*sin(pi/4)}}} = i;

        b)  {{{cis(pi/4 + 2pi/3)}}} = {{{cis(pi/4)*cis(2pi/3)}}} = {{{i*cis(2pi/3)}}};

        c)  {{{cis(pi/4 + 4pi/3)}}} = {{{cis(pi/4)*cis(4pi/3)}}} = {{{i*cis(4pi/3)}}}.

        Therefore, the sum of the three cube roots is

        {{{i + i*cis(2pi/3) + i*cis(2pi/3)}}} = {{{i*(1+cis(2pi/3)+cis(4pi/3))}}} = 0,

        since {{{1+cis(2pi/3)+cis(4pi/3)}}} = 0,  as you can easily check.
</pre>


2.  Same technique and the same logic works for the fourth roots of {{{ -1/2- (sqrt (3) /2)i }}}.


<pre>
    You can check it on your own as an exercise for you.
</pre>


There is a bunch of lessons on complex numbers

&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/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>

in this site.


Also, 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 "<U>Complex numbers</U>".