Properties of Cube Roots of Unity

Properties of Cube Roots of Unity

## Formula

$$\omega^3=1$$

$$\omega^4=\omega$$

$$\omega^5=\omega^2$$

$$\omega^6=1$$

Therefore:

$$\omega^{3n}=1$$

$$\omega^{3n+1}=\omega$$

$$\omega^{3n+2}=\omega^2$$

## Important Identities

$$\omega+\omega^2=-1$$

$$\omega-\omega^2=i\sqrt3$$

$$\omega^2-\omega=-i\sqrt3$$

$$\omega\omega^2=1$$

## Usage

- Used to reduce high powers involving $\omega$ and simplify cyclic expressions.