Complex numbers, finding smallest exponent

Question

$$ z= \frac{\sqrt 3 -i}{1+ \sqrt3 i}$$ I need to find smallest exponent $n>2018$, such as $z^n$ will be a number with real part equal to $0$ and imaginary part of number will be negative. I calculated that $z=-i$ but I don't know how to write a equation that covers all conditions. It doesn't have to be an exact answer, I am asking more about how do I solve this kind of a problem effectively

Answer 1

Note that\begin{align}\require{cancel}z&=\frac{(-i)\cancel{\left(1+\sqrt3\,i\right)}\left(1-\sqrt3\,i\right)}{\cancel{1+\sqrt3\,i}}\\&=-\sqrt3-i\\&=2\left(-\frac{\sqrt3}2-\frac12i\right)\\&=2\left(\cos\left(\frac{7\pi}6\right)+\sin\left(\frac{7\pi}6\right)i\right).\end{align}Can you take it from here?