Question:
If $k, l, r, s \in \mathbb{Z^+} \text{ and } k^r=l^s$ prove that $k \text{ and } l$ are powers of some positive integer
What I Tried:
I attempted to assume that $k=m^q \text{ and } l=n^p$ where $p, q \in \mathbb{Z^+}$. Now I need to prove that either $m=n$ or $m$ is a power of $n$ (could be $n$ as power of $m$). Now we can say that: $$m^{qr}=n^{ps}$$ I am unsure if this step is necessary but I could rename $qr$ into $a$ and $ns$ into $b$ $$\therefore m^a=n^b$$ Now maybe I tried to repeat what I did earlier but that didn't help, so I was wondering if there is a different step or approach from here. Thank you anyways.
Your solution is wrong; you cannot immediately assume they are perfect powers without further justification. Instead, since we can just take roots of both sides if necessary,we may assume $\gcd(r,s)=1$. Further we have $k^{r/s}=l\in\mathbb Z$, hence $k^r$ is a perfect $s$th power. Since $\gcd(r,s)=1$ this means $k$ is a perfect $s$th power too. Hence let $k=k_1^s$, where $k_1\in\mathbb Z$. By the exact same logic we also have $l=l_1^r$. Substituting inside, we get $k_1^{rs}=l_1^{rs}$, which implies $k_1=l_1$.