Question

If r, s, and t are constants such that %20z%5E%7B3t%2B1%7D%7D%7Bx%5E%7B2r%7D%5Ccdot%20y%5E%7Bs-4%7D%5Ccdot%20z%5E%7B2t-3%7D%7D%3Dxyz%20" id="TexFormula1" title=" \frac{x^{r-2}\cdot y^{2s}\cdot z^{3t+1}}{x^{2r}\cdot y^{s-4}\cdot z^{2t-3}}=xyz " alt=" \frac{x^{r-2}\cdot y^{2s}\cdot z^{3t+1}}{x^{2r}\cdot y^{s-4}\cdot z^{2t-3}}=xyz " align="absmiddle" class="latex-formula"> for all non-zero x, y, and z, then solve for $r^s\cdot t$. Express your answer as a fraction.

Answer 1

$\dfrac{x^{r-2}y^{2s}z^{3t+1}}{x^{2r}y^{s-4}z^{2t-3}}=x^{-r-2}y^{s+4}z^{t+4}=xyz$

$\implies\begin{cases}-r-2=1\\s+4=1\\t+4=1\end{cases}\implies r=s=t=-3$

$\implies r^st=(-3)^{-3}(-3)=(-3)^{-2}=\dfrac1{(-3)^2}=\dfrac19$