SOLUTION: finding the nth root of a complex number (16i)^1/4

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Question 1178763: finding the nth root of a complex number
(16i)^1/4

Answer by greenestamps(13200) About Me  (Show Source):
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16i+=+0%2B16i+=+16cis%28pi%2F2%29

Use deMoivre's Theorem on the last form to find the 4th roots

(1) Find the "primary" root. To find the nth root of a number in a*cis(theta) form, take the nth root of a, and divide the angle theta by n.



(2) Find the other roots. The n n-th roots of a number all have the same magnitude, and they are distributed around the complex plane in intervals of (2pi)/n.

2pi%2F4+=+pi%2F2

The 4th roots are at intervals of pi/2 in the complex plane. Starting with the "primary" root of 2*cis(pi/8), the four 4th roots of 16i are

2cis(pi/8)
2cis(5pi/8)
2cis(9pi/8)
2cis(13pi/8)