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.
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 = 163/4 = 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