Question

Use Demoiure's Theorem to find (-2-2√3i)³

Answer 1

Let z = -2 - 2√3 i.

Find the modulus of z :

|z| = √((-2)² + (-2√3)²) = √16 = 4

Find the argument of z (note that z lies in the third quadrant):

arg(z) = arctan(2√3/2) - π = arctan(√3) - π = -2π/3

Then in polar form, we have

`z = -2 - 2\sqrt 3\, i = 4 \exp\left(-i\frac{2\pi}3\right)`

By DeMoivre's theorem, the cube of z is

`z^3 = \left(4 \exp\left(-i\frac{2\pi}3\right)\right)^3 = 4^3 \exp(-i 2\pi) = \boxed{64}`