SOLUTION: Find all the sixth roots of (12+5i).

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Question 421149: Find all the sixth roots of (12+5i).
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
z%5E6+=+12%2B5i
z%5E6+=+13%2812%2F13+%2B+5i%2F13%29
==> z+=+root%286%2C+13%29%2812%2F13+%2B+5i%2F13%29%5E%281%2F6%29
Let theta be the angle in quadrant 1 such that cos+%28theta%29+=+12%2F13 and sin%28theta%29+=+5%2F13, or tan+%28theta%29+=+5%2F12. Then
z+=+root%286%2C+13%29%28cos%28theta%29+%2B+i%2Asin%28theta%29%29%5E%281%2F6%29
<==> z+=+root%286%2C+13%29%28cos%28theta%2F6%29+%2B+i%2Asin%28theta%2F6%29%29 by de Moivre's Theorem.
This is the principal 6th root. The other roots are given by
, letting n = 1,2,3,4,5.