Finding roots of complex number

The problem is specific as an example from hw. But it is more the concept/process I could use clarification on. Given a complex number

$$\Big(\frac{-2}{1-i\sqrt3}\Big)^{\frac{1}{4}}$$

Find all possible roots. I know the method is to change into the exponential form, solving for magnitude (r) and theta. Which I did and got $e^{i4\pi/3}$. Do I then multiply theta by $4$, or $1/4$ and then add $\pi/2$ ? By either multiplying or dividing I get the same 4 roots, only with different starting points. But, if all I want are the roots, does it matter what the starting point is?

Answer

In general, we can write

$$z^c=e^{c\log(z)}$$

where the complex logarithm is the multi-valued function and can be written as

$$\log(z)=\log(|z|)+i(\arg(z)+2n\pi)$$

for all integer values of $n$.

For the specific problem in the OP, $z=\frac{-2}{1-i\sqrt 3}=\frac{2e^{-i\pi}}{2e^{-i\pi/3}}=e^{-i2\pi/3}$ and $c=1/4$. Therefore, we have

$$\begin{align}
\left(\frac{-2}{1-i\sqrt 3}\right)^{1/4}&=e^{\frac14\log(1)+i\frac14(-2\pi/3+2n\pi)}\\\\

&e^{i(-\pi/6+n\pi/2)}\\\\
&=(i)^ne^{-i\pi/6}\\\\
&=
\begin{cases}
ie^{-i\pi/6}&,n=1\\\\
-1e^{-i\pi/6}&,n=2\\\\
-ie^{-i\pi/6}&,n=3\\\\
e^{-i\pi/6}&,n=4
\end{cases}
\end{align}$$