Real numbers $a$ and $b$ satisfy the equations $3^a=81^{b+2}$ and $125^b=5^{a-3}$.  What is $ab$?
The given equations are equivalent, respectively, to \[
3^a=3^{4(b+2)}\quad\text{and}\quad 5^{3b}=5^{a-3}.
\] Therefore $a=4(b+2)$ and $3b=a-3$.  The solution of this system is $a=-12$ and $b=-5$, so $ab=\boxed{60}$.