SOLUTION: Find the roots and express your answer in standard/ rectangular form. 1. (-2+2i)^1/3

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Question 1178751: Find the roots and express your answer in standard/ rectangular form.
1. (-2+2i)^1/3
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Write the given complex number in the form A cis(B) = A*(cos(B)+i*sin(B)).

-2%2B2i+=+2%2Asqrt%282%29cis%28135%29

Use deMoivre's Theorem to find a power (integer or fraction) of the complex number: To find the n-th power of the complex number, raise the magnitude A to the nth power and multiply the angle by n.

%282%2Asqrt%282%29%29%5E%281%2F3%29+=+sqrt%282%29
135%2A%281%2F3%29+=+45

The "primary" root is sqrt%282%29%2Acis%2845%29

The n n-th roots of a complex number are distributed around the complex plane at intervals of 360/n degrees. So

ANSWER: The three cube roots of -2+2i are
sqrt%282%29%2Acis%2845%29
sqrt%282%29%2Acis%28165%29
sqrt%282%29%2Acis%28285%29

Convert to rectangular form using
cos%2845%29+=+sin%2845%29+=+sqrt%282%29%2F2
cos%28165%29+=+sin%28285%29+=+-%28%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29
sin%28165%29+=+cos%28285%29+=+%28sqrt%286%29-sqrt%282%29%29%2F4