Find the nth roots of the complex number for the specified value of n.

Question

Find the nth roots of the complex number for the specified value of n.
2-2i, n=4

Answer 1

Using the complex number z = 2 - 2i and we want the roots for n = 4, consider writing z using Euler notation...

(x+iy) = 2 - 2i = re^iθ

r = sqrt(2² + 2²) = 2√2

θ = arctan(y/x) = arctan(-2/2) = 3π/2

2 - 2i = 2√2 e^3πi/2

The fourth roots are easy to compute now...

(2√2)^(1/4) = (√8)^(1/4) = 8^(1/8)

e^{(3πi/2)(1/4)} = e^3πi/8

The fourth roots are 8(1/8) * e^{(3πi/8) + (n/4)} with n ranging from 0 .. 3