I dont understand where to begin, or how to approach this question.
it asks:
find all the roots of:
$$(1 + \sqrt{3}i) ^{1/2}$$
should I put it into polar form first?
$$z = re^{ix}$$
what throws me off on this question is that it is raised to the 1/2 I wrote it as
$$z = 2\left(\cos\frac{\pi}{3}+ i\sin\frac{\pi}{3}\right)$$
Recall that if you take the $n$th root you should be looking for $n$ solutions.
$$\left(re^{ix}\right)^{\frac{1}{2}} = r^{1/2}e^{ix/2}$$
This gives us one equation. Additionally, recall that $e^{ix} = e^{ix+2i\pi n}, n \in \mathbb{Z}$. We can find another solution by setting $n = 1$. (Why can't we find a third by setting $n = 2$?)
$$\left(re^{ix}\right)^{\frac{1}{2}} = r^{1/2}e^{ix/2+2i\pi/2} = r^{1/2}e^{(ix + i\pi)/2}$$
In your equation, $r = 2$ and $x = \frac{\pi}{3}$.
