De Moivre's Formula/Exponential Form

Theorem

$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$

Proof

 $\displaystyle \paren {r e^{i \theta} }^\omega$ $=$ $\displaystyle \paren {r \paren {\cos x + i \sin x} }^\omega$ Definition of Exponential Form of Complex Number $\displaystyle$ $=$ $\displaystyle r^\omega \paren {\cos \omega x + i \omega \sin x}$ De Moivre's Formula $\displaystyle$ $=$ $\displaystyle r^\omega e^{i \omega \theta}$ Definition of Exponential Form of Complex Number

$\blacksquare$