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# r s and t constants

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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.

Oct 2, 2020

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It looks a lot scarier than it really is

Just simplify the LHS

for instance the x term will just be    $$x^{r-2-(2r)}=x^{-r-2}$$

Once you have done that then you can equate the powers.

so for the x term

$$x^{-r-2}=x^1\\ so\\ -r-2=1\\ r=-3$$

you can find s and t the same way.

Then you can get the answer.