Roman C. answered • 11/28/12

Masters of Education Graduate with Mathematics Expertise

One way to write a complex number a + bi is

r cos θ + i r sin θ

This number is obtained by rotating r+0i counterclockwise by angle θ in the complex plane.

r = |a+bi| = √(a^{2}+b^{2}) is called the modulus. While θ is called the argument.

De Moivre's theorem says that when you multiply two complex numbers, you take the product of their moduli and the sum of their arguments. Also, when raising a complex number to the power of x, you raise the modulus to the power of x and multiply the argument by x.

For example:

8 - 8i√3 = 16 cos (-π/3) + i 16 sin (-π/3)

has r = 16 and θ = -π/3

Therefore (8 - 8i√3)^{3/4} has

r = 16^{3/4} = 8 and θ = (3/4) * -π/3 = -π/4

Thus (8 - 8i√3)^{3/4} = 8 cos (-π/4) + 8i sin (-π/4) = 4√2 - 4i√2

Robert J.

Attn: You have 4 different solutions for 4th root of a complex number.

11/29/12