For $s>0$, define $F_s:\mathbb{R}^n\to\mathbb{R}^n$, $x\mapsto |x|^{s-1}x$ which is a homeomorphism and a diffeomorphism if and only if $s=1$. Now for $s,t>0$ with $s\neq t$ and $s,t\neq 1$, I would like to show that at least one of $F_t\circ F_s^{-1}$ or $F_s\circ F_t^{-1}$ is not smooth (here smooth means $C^\infty$). I realize that this should be fairly straight forward, but I am not sure how to come up with a short proof/explanation for it. I thought of trying to find $F_s^{-1}$ explicitly but I am wondering if there is some straight-forward way of showing this.
For background, I am trying to prove that a collection of smooth structures are not compatible (differential geometry).