Does $\left\{\frac{p^a}{q^b}:p,q\in\mathbb{Z}\right\}=\left\{\frac{j^{\gcd(a,b)}}{k^{\gcd(a,b)}}:j,k\in\mathbb{Z}\right\}$ for $a,b\in\mathbb{N}$?
Yes, it is true.
The containment $\subseteq$ is easy to show.
For $\supseteq$, take any $a,b,i,j$. Let $x_1,x_2,y_1,y_2$ be such that $ax_1+by_1=ax_2+by_2=d=\operatorname{gcd}(a,b)$ and $x_1,y_2\geq 0\geq y_1,x_2$. Take $p=i^{x_1}j^{-x_2}$, $q=i^{-y_1}j^{y_2}$. Then $p^aq^{-b}=i^{ax_1+by_1}j^{-ax_2-by_2}=i^dj^{-d}$ and you are done.