smc

Let $(xy{)}^2=a$. The expression given is equal to $a^2-10a+9=0$, which can be factored as $(a-9)(a-1)=0$. Thus, we have $a=(xy{)}^2=9$ and $a=(xy{)}^2=1$. Because $x$ and $y$ are positive, we can eliminate the possibilities where $xy$ is negative. From here it is easy to see that the only integral pairs of $x$ and $y$ are $(3,1),(1,3)$, and $(1,1)$. The answer is $\boxed{3}$.