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

Not by definition, just check that
$$\omega=\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i=(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))=e^{i\frac{\pi}{3}}$$
If you make $\omega^3$, then you have $e^{i\frac{\pi}{3}\cdot 3}=e^{i\pi}=-1$.
So $\omega^5=\omega^3\omega^2=-\omega^2$ is your mistake
