jmc

First, we should solve for $r$, $s$, and $t$. From what is given, we know that $\frac{x^{r-2}}{x^{2r}}=x$, $\frac{y^{2s}}{y^{s-4}}=y$, and $\frac{z^{3t+1}}{z^{2t-3}}=z$. Solving for r, s, and t we have: $$\begin{array}{r}r-2=2r+1\Rightarrow r=-3\\ 2s=s-4+1\Rightarrow s=-3\\ 3t+1=2t-3+1\Rightarrow t=-3\end{array}$$Solving for $r^s\cdot t$, we have $(-3{)}^{-3}\cdot -3=\frac{-1}{27}\cdot -3=\boxed{\frac{1}{9}}$.