Good day, everybody!
I was handed this problem by a friend. Prove that $|z|=1$ if $z^n=1$.
$z$, in this case, is a complex number.
It should be very easy, but I'm missing something....
Could someone please provide a step by step solution? So far, I've tried taking the magnitude of both sides of the given equation, but I don't know what to do after...
With the hint of the first comment you have $$1=z^n\implies 1=|z^n|=|z|^n$$
Since $|z|$ is a non-negative real number, it must be $1$, because $1$ is the only solution of $x^n=1$ in non-negative real numbers. To see that consider the cases $0\le x<1$ and $x>1$.