Coprime numbers and exponents

Prime factorization. $a=\prod p^{u_p}$, $b=\prod p^{v_p}$. Since $\gcd(a,b)=1$, for each prime $p$ at most one of $u_p,v_p$ is positive.

Equality. $a^m=b^n$ means $mu_p=nv_p$ for all $p$.

Conclusion. If $u_p>0$, then $v_p=0$ and $mu_p=0$, contradiction. So $u_p=0$ for all $p$, i.e. $a=1$. Similarly $b=1$.

Correction: The statement as written is trivial if interpreted literally. The correct nontrivial statement is: $a^m=b^n$ implies both are powers of a common integer. Indeed, with $d=\gcd(m,n)$ and $m=d m'$, $n=d n'$, we have $a^{m'}=b^{n'}$, and since $\gcd(m',n')=1$ we get $a=c^{n'}$, $b=c^{m'}$ for some integer $c$.

Conclusion. $a$ and $b$ are both powers of a common integer.