using demoivre's theorem, what is the value of (8-8iv 3)^3/4

Question

I am having trouble understanding demoivre's theorem. I am not quite grasping the concept. Please help with this homework problem.

Answer 1

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| = √(a²+b²) 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

 

Answer 2

You start with 8-8i√3 = 16cis(-pi/3 + 2n*pi)

(8-8i√3)^(3/4)

= [16cis(-pi/3 + 2n*pi)]^(3/4)

= 8cis(-pi/4 + 3n*pi/2), n = 0, 1, 2, 3

Answer in rectangular form: 4√2 -  4i√2, -4√2 - 4i√2, -4√2 + 4i√2, 4√2 + 4i√2.