I want to prove if $|z|=S$, then $$S^m=|z|^m=|z^m|$$.
Work done so far:
The first one is quite easy to prove: if $|z|=S$, then raising both sides to the power of $m$ will give us $s^m=|z|^m$.
I am struggling with the second part; any ideas on how to prove it?
Edit: After a helpful hint this is a proof I came up with:
Let $z=re^{i\theta}$, then $z^m=r^me^{mi\theta}$. But, $|z|=r$ so $|z^m|=r^m$ Furthermore, since, $|z|=r$, $|z|^m=r^m$
Therefore, $S^m=|z|^m=|z^m|$.
How is this proved?
You already have it there. It'd seem that your problem is recognizing it. My recomendation would be that you write down your proof carefully annotating every derivation step with an arrow ($\Rightarrow$). This arrow means
if the left-hand-side is true, then we can conclude the right-hand-size, and the reason for that to be true must be easy to understand -otherwise it'd require a prove on it's own.In the end the full proof can be read as a single, big
if-thenstatement: $$|z|=\sqrt{z\bar{z}}\,\Rightarrow\,|z|^n\,=\,\sqrt{(z\bar{z})^n}\,\Rightarrow\,|z|^n\,=\,|z^n|$$