De Moivre's Theorem

De Moivre's theorem extends the polar-form multiplication rule to whole-number powers:

$\big[r(\cos\theta + i\sin\theta)\big]^n = r^n(\cos n\theta + i\sin n\theta)$

It turns "raise a complex number to a power" — normally a long binomial expansion — into just scaling the modulus and multiplying the angle.

Example: $(\cos 30^\circ + i\sin 30^\circ)^3 = \cos 90^\circ + i\sin 90^\circ = i$.

It's also the standard tool for finding roots of unity (see the Cube/Fourth Roots of Unity topics) by running the formula in reverse with fractional $n$.