Finding the values of this complex number

Question

I need to find the values of $(1 - i\sqrt{3})^i$ in terms of $a + bi$. First I found $r$ which was $2$ and $\theta$ which was $5\pi/3$ or $-\pi/3$. So $z = 2e^{i\theta}$ all to the i power but I'm lost on what to do from here.

Answer 1

$$(2e^{i\theta})^i = 2^i e^{-\theta} = e^{i\log 2} e^{-\theta}$$ and use $ e^{i\theta} = \cos \theta + i\sin \theta$ from here...

Answer 2

Since $1-i\sqrt{3}=2\exp\frac{5\pi i}{3}=\exp\left(\ln 2+\frac{\pi i(6n+5)}{3}\right)$ for $n\in\Bbb Z$, $(1-i\sqrt{3})^i$ is multi-valued with possible values $\exp\left(i\ln 2-\frac{\pi(6n+5)}{3}\right)$.