If $z$ is an $n$th root of unity, prove that $1/z$ is an $n$th root of unity

Question

I'm not sure if how I'm going to prove this to be correct:

Since $z$ is an $n$th root of unity, it means $z^n = 1$

For $1/z$ to be an nth root of unity, lets take it to the power of $n$,

$(1/z)^n$, and so $1/(z^n) = 1/1 = 1$, hence, $1/z$ is an $n$th root of unity ?

Answer 1

Nicely done! That's exactly the most straightforward way to prove it.