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If r, s, and t are constants such that [tex] \frac{x^{r-2}\cdot y^{2s}\cdot z^{3t+1}}{x^{2r}\cdot y^{s-4}\cdot z^{2t-3}}=xyz [/tex] for all non-zero x, y, and z, then solve for [tex] r^s\cdot t [/tex]. Express your answer as a fraction.

Respuesta :

LammettHash

[tex]\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[/tex]

[tex]\implies\begin{cases}-r-2=1\\s+4=1\\t+4=1\end{cases}\implies r=s=t=-3[/tex]

[tex]\implies r^st=(-3)^{-3}(-3)=(-3)^{-2}=\dfrac1{(-3)^2}=\dfrac19[/tex]

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