Exponentiating roots of unity

Question

When given some number some complex $2n^{th}$ root of unity, $z$, how does one evaluate something such as $z^{2m}$ (with $m=kn$)? I would take $z^{2m}=(z^{2n})^k=1^k=1$, but I don't know if this is always correct.

Answer 1

Yes, that is correct. Raising complex numbers to integer powers follow all the well-known power laws, and in particular the ones you use here.