Large polynomial power of a complex number

Question

I need to work out whether $(\sqrt3-i)^{2014}$ is real or not?

So far I have $|z|=2, \ \arg z=-\pi/6$

Then I put into polar form: $z=2e^{-i\pi/6}$

Then raised to the power of 2014: $z^{2014}=2^{2014}e^{-2014i\pi/6}$

But now since 2014/6 does not divide, I am not sure how to reduce further? I was thinking of using modular arithmetic so $2014=(335*6)+4$?

Any help would be greatly appreciated!

Answer 1

$e^{i\theta}$ is real if and only if $\theta$ is an integer multiple of $\pi$. Is $-2014/6$ an integer?