De Moivre's Formula/Rational Index

Theorem
Let $z \in \C$ be a complex number expressed in complex form:
 * $z = r \left({\cos x + i \sin x}\right)$

Then:
 * $\forall p \in \Q: \left({r \left({\cos x + i \sin x}\right)}\right)^p = r^p \left({\cos \left({p x}\right) + i \sin \left({p x}\right)}\right)$

Also defined as
This result is also often presented in the simpler form:


 * $\forall p \in \Q: \left({\cos x + i \sin x}\right)^p = \cos \left({p x}\right) + i \sin \left({p x}\right)$

Also known as
De Moivre's Theorem.