Question

If r, s, and t are constants such that
`(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.

Answer 1

`(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)=\frac1{(-3)^2}=\frac19`