De Moivre's Formula/Exponential Form

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Theorem

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


Proof

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

$\blacksquare$


Sources