Question

Compute the indicated power by using DeMoivre’s theorem:

Answer 1

`2+2i=2\sqrt2e^(i\pi/4)`

`3-3i=3\sqrt2e^(-i\pi/4)`

`\implies(2+2i)/(3-3i)=\frac23e^(i\pi/2)`

By DeMoivre's theorem,

`\left((2+2i)/(3-3i)\right)^5=\left(\frac23\right)^5e^(i5\pi/2)=(32)/(243)e^(i\pi/2)=(32i)/(243)`

Just to confirm:

`(2+2i)/(3-3i)\cdot(3+3i)/(3+3i)=(12i)/(18)=\frac{2i}3\implies\left((2+2i)/(3-3i)\right)^5=(32i)/(243)`