# How do you use De Moivre’s Theorem to find the powers of complex numbers in polar form?

Dec 3, 2014

If the complex number $z$ is

$z = r \left(\cos \theta + i \sin \theta\right)$,

then ${z}^{n}$ can be written as

${z}^{n} = {\left[r \left(\cos \theta + i \sin \theta\right)\right]}^{n} = {r}^{n} {\left[\cos \theta + i \sin \theta\right]}^{n}$

by De Mivre's Theorem,

$= {r}^{n} \left[\cos \left(n \theta\right) + i \sin \left(n \theta\right)\right]$

I hope that this was helpful.