Im answering the following question: Prove that if $z=u$ is a solution of the equation $z^n=w$, then the other solutions have the form $u \rho ^j$ for $j=1,2,...,n-1$.
The only thing I need to know if my answer is right is if for $z, w \in \mathbb{C}$ we have that $(zw)^{n}=z^{n}w^{n}$ for $n\in \mathbb{N}$. Im not studying this as a complex analysis course so I just need to know if this is true or give me a glimpse if this is true. Im not looking for a fancy proof but a little sketch of the proof or link to the proof of this will be helpful. Thanks for helping
Thanks, I already did a proof by induction that goes as follows: If we suppose $(zw)^{k-1}=z^{k-1}w^{k-1}$ then $$(zw)^{k}=(zw)^{k-1}(zw)=z^{k-1}w^{k-1}(zw)=z^{k}w^{k}.$$ The last step by the fact $\mathbb{C}$ is a commutative ring.