SOLUTION: determine all the indicated roots and represent them graphically : a) (4*√2+4*√2i)^1/3

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Question 1104069: determine all the indicated roots and represent them graphically :
a) (4*√2+4*√2i)^1/3
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Here is a representation of 4sqrt%282%29%2Bi4sqrt%282%29
4sqrt%282%29%2Bi4sqrt%282%29=r%28cos%28theta%29%2Bi%2Asin%28theta%29%29
Obviously, tan%28theta%29=4sqrt%282%29%2F4sqrt%282%29=1 , so theta=pi%2F4 .
In degrees, theta=45%5Eo .
r=sqrt%28%284sqrt%282%29%29%5E2%2B%284sqrt%282%29%29%5E2%29=sqrt%2832%2B32%29=sqrt%2864%29=8
So,
4sqrt%282%29%2Bi4sqrt%282%29=8%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29 .
There are 3 complex values for %288%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29%29%5E%281%2F3%29 .
They are complex numbers of the form r%28cos%28theta%29%2Bi%2Asin%28theta%29%29
such that .
So, it will be r%5E3=8 and 3theta=pi%2F4%2Bk%282pi%29 for %22k+=+0+%2C+1+%2C+2%22 .
In degrees, 3theta=45%5Eo%2Bk360%5Eo .
That means r=2 and
system%28theta=pi%2F12%2Ctheta=3pi%2F4%2Ctheta=17pi%2F12%29 .
In degrees, .
The answers could be written as
green%28Z%5B1%5D%29=2cos%2815%5Eo%29%2Bi%2Asin%2815%5Eo%29 ,red%28Z%5B2%5D%29=2cos%28135%5Eo%29%2Bi%2Asin%28135%5Eo%29 , and blue%28Z%5B3%5D%29=2cos%28255%5Eo%29%2Bi%2Asin%28255%5Eo%29
(or you could express the angles in radians).
They can be represented as