De Moivre's theorem

De Moivre's theorem is a shortcut for raising a complex number to a power when the number is written in polar form. Instead of multiplying the number by itself again and again — which gets painful fast — you simply raise its modulus to the power and multiply its argument by that power. Powers become almost arithmetic.

Precisely, (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for any integer n. With a modulus r included, [r(cos θ + i sin θ)]^n = r^n (cos(nθ) + i sin(nθ)). The intuition matches the multiplication rule: each time you multiply a number by itself, you add its argument and multiply its modulus, so raising to the n-th power adds the argument n times and raises the modulus to the n-th power.

The same formula, run in reverse, finds roots. To take an n-th root you divide the argument by n and take the n-th root of the modulus, and because angles repeat every full turn, you get n different roots evenly spaced around a circle. This is the engine behind the roots of unity.