If $z$ is a non-zero complex number and $p,q,m,n$ are positive jntegers where $\frac{p}{q}=\frac{m}{n}$ with $gcd(m,n)=1$, then

Question

If $z$ is a non-zero complex number and $p,q,m,n$ are positive jntegers where $\frac{p}{q}=\frac{m}{n}$ with $gcd(m,n)=1$, then $z^{\frac{p}{q}}=z^{\frac{m}{n}}$.

The proof given in the book is as follows:

Here, $np=qm$. Let $w=z^{p/q}$. Then $w=(z^{1/q})^p$, by definition. $w^n={\{z^{(1/q)p}\}}^n=(z^{1/q})^{qm}={\{(z^{1/q})^{q}\}}^m=z^m$. Therefore, $w=(z^m)^{1/n}=(z^{1/n})^m=z^{m/n}$, since $gcd(m,n)=1$.

However , I am not getting how they are writing "$w=(z^m)^{1/n}=(z^{1/n})^m=z^{m/n}$"? I mean if there is a expression say $z^p=k^q$ then $k={\{z^p}\}^{1/q}$, i.e k will have one of the q distinct values of $z^p$. Can one write $k=z^n\implies k^{1/n}={\{z^{n}\}}^{1/n}=z$? so what about k ? There is a theorem if $z$ is a complex no. then there are $n$ distinct values of $z^{1/n}$. So there should be $n$ distinct values of $k^{1/n}$? So can we write $k^{1/n}=z$? Same goes for $w$...How is that possible?

Also, there is a similar type of post I made (not exactly similar but related) in the community If $z$ is a non-zero complex number and $m,n\in\mathbb{Z^+}$ and $gcd(m,n)=1$,then $(z^{\frac{1}{n}})^m=(z^m)^{\frac{1}{n}}$. . But no answers there as of yet ...[I am not aware whether we can do this , to bring up an older post in a current post]...but that query wasn't quite solved either...and I think these two are quite fundamental questions I asked..and in many books (I have seen) none explains these things in details ...I don't know whether it's obvious or not...but if obvious then how? Can someone please explain it ...I am not quite getting it...

Answer 1

Complex power is by definition, $z^w := \exp\left(w\log z\right)$, where the $\log z$ is a multi-valued function. Using this definition, the conclution is immediate. More precisely, one value of $z^{p/q}$ is a value of $z^{m/n}$ and vice versa.