Please show how step by step (with explaination) to obtain fifth root of 4+3i.
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Basically we have,
z^5 = 4 + 3i
Say, w = 4 + 3i = rcisθ
r = (4^2 + 3^2)^0.5
tan θ = 3/4, so θ = arctan (0.75), which in degrees is approximately equal to pi/10, so
w = 5 cis (pi/10 + 2k pi), since we're going to be dealing with the 5th root here, k can be equal to 0,1,2,3 or 4.
z^5 = 5 cis (pi /10 + 2k pi)
z = [5 cis (pi/10 + 2k pi)^(1/5)]
= 5^1/5 cis (pi/50 + 2k pi / 5)
First root is when k = 0, second root is when k = 1 and so on...
so the fifth root is when k = 4, so we have:
5^(1/5) cis (pi/50 + 8k pi / 5)
This isn't the exact answer because of the approximation I made earlier, but it's close.