$\omega = \frac{1 + \sqrt3 i}{2}$ , $ \omega^5 = ? $ 

$\omega = \frac{1 + \sqrt3 i}{2} $, $ \omega^5 = ? $


$\omega^3 = 1$ by definition?
So, $\omega^5 = \omega^2$
But why do i get wrong answer?

Because $\omega=\cos\left(\frac\pi3\right)+i\sin\left(\frac\pi3\right)$ and therefore it is not true that $\omega^3=1$. Actually, $\omega^3=-1$. And$$\omega^5=-\omega^2=-\cos\left(\frac{2\pi}3\right)-i\sin\left(\frac{2\pi}3\right)=\frac{1-i\sqrt3}2.$$
