I have been trying to write $\omega^2$ in Euler form, first $\omega^2 = \frac{-1}{2}-\frac{\sqrt{3}i}{2} ,$ hence $|\omega^2| = 1$ and $\arg(\omega^2) = -\pi + \frac{\pi}{3}$ as $\omega ^2$ lies in the fourth quadrant which gives $\omega^2 = e^{\frac{-2i\pi}{3}}$.
But using de Moivre's formula, I derived that the cube roots for unity is $1$, $\exp \left(\frac{2i\pi}{3} \right)$ and $\exp \left( \frac{4i\pi}{3} \right)$. I am getting the value of $\omega = e^{\frac{2i\pi}{3}} $ but I am not sure how $e^{\frac{4i\pi}{3}}=e^{\frac{-2i\pi}{3}}$? Please explain your answer.
Thanks,