Roots of unity has length 1?

Question

It seems to be the case that if $z\in\mathbb{C}$ is a root of unity, i.e. $z^k=1$ for some $k\in\mathbb{N}$, then $z$ lies on the unit circle in the complex plane, in other words $|z|=1$.

Let $z=a+bi$ be such that $z^k=1$, how can I show that $|z|=\sqrt{a^2+b^2}=1$?

Answer 1

Hint: Suppose $\lvert z\rvert \neq 1$. What can you say about $\lvert z^k\rvert$?

Answer 2

Hint:

As the modulus of a complex number is a multiplicative function, the modulus of a $k$-th root of unity is a positive real root of unity. Do you know many of them?