# Math Help - fifth root

1. ## fifth root

Please show how step by step (with explaination) to obtain fifth root of 4+3i.

2. Originally Posted by rajuk
Please show how step by step (with explaination) to obtain fifth root of 4+3i.
The general technique is shown here: http://www.mathhelpforum.com/math-he...solutions.html

You will find that this sort of question has been solved several times in the forums. The Search tool will probably find more threads. (And Google will find even more).

If you need more help, please show your work and state where you are stuck.

3. Basically we have,
z^5 = 4 + 3i
Say, w = 4 + 3i = rcisθ
r = (4^2 + 3^2)^0.5
= 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.