Let $x,y,a,b \in \mathbb{Z}>1 $and $\gcd(x,b)=1,$ $y^{ax}=x^2b$, I cannot find any integral solution.
What I have done so far: I assume there must be 2 coprime integers $c, d>1$ such that $$y=c^2d$$ $$x=c^{ax}$$ $$b=d^{ax}$$ And conclude that there is no integer $x>1$ such that $x=c^{ax}$. Am I correct?Please any insight will be greatly appreciated.