Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Question

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$

I'm in Algebra two and I feel like there are certain types of math I haven't learned that need to be used in this problem or maybe not. Please show the steps for solving it.

Answer 1

HINT: we get from the first equation $$\frac{3^{3(x+y)}}{3^{2xy}}=3^3$$ from here we obtain $$3^{3(x+y)-2xy}=3^3$$ and finaly $$3(x+y)-2xy=3$$ analogously from the second equation $$4xy-3(x+y)=9$$