### Instruction: If $r$, $s$, and $t$ are constants such that $\frac{x^{r-2}\cdot y^{2s}\cdot z^{3t+1}}{x^{2r}\cdot y^{s-4}\cdot z^{2t-3}}=xyz$ for all non-zero $x$, $y$, and $z$, then solve for $r^s\cdot t$. Express your answer as a fraction.### Response: This problem involves exponents and fractions, so I want to simplify the expression as much as possible.
I notice that I can use the quotient rule for exponents to combine the exponents of the same base in the numerator and denominator.
For example, $\frac{x^{r-2}}{x^{2r}}=x^{r-2-2r}=x^{-r-2}$.
Similarly, $\frac{y^{2s}}{y^{s-4}}=y^{2s-s+4}=y^{s+4}$ and $\frac{z^{3t+1}}{z^{2t-3}}=z^{3t+1-2t+3}=z^{t+4}$.
So, the expression becomes $x^{-r-2}\cdot y^{s+4}\cdot z^{t+4}=xyz$.
Now, I want to compare the exponents of the same base on both sides of the equation.
For $x$, I have $-r-2=1$, so $r=-3$.
For $y$, I have $s+4=1$, so $s=-3$.
For $z$, I have $t+4=1$, so $t=-3$.
Therefore, the values of $r$, $s$, and $t$ are all $-3$.
To find $r^s\cdot t$, I just plug in these values.