Question 1046939
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Determine the fourth roots of -16
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<pre>
In the complex plane, -16 = {{{2^4*(cos(pi) + i*sin(pi))}}}.

The modulus of -16 is {{{2^4}}}, the argument is {{{pi}}}.

Therefore, according to the general theory, the fourth roots of -16 are four complex numbers


   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>

For this "general theory" see the lesson 

&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>

in this site.


There is a bunch of my 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>

in this site.