Negative exponent of exponent in cyclic group of prime power order.

Question

Given a cyclic group $G$ of order $|G| = p^s$, we have that $x^{p^s} = e$ for all $x\in G$. Consider then $x^{p^{s-t}}$: $$ x^{p^{s-t}} = x^{p^sp^{-t}} = \left(x^{p^s}\right)^{p^{-t}} = e^{p^{-t}} = e $$ This result is of course invalid ($s=t$ implies $x=e$ for all $x$), but what step is false? And is there a way to interpret $x^{p^{-t}}$?