jmc

Since $81=3^4$, then $3=81^{1/4}$. Comparing exponents, it follows that we have the system of equations $$\begin{array}{rl}x+y& =4\\ x-y& =1/4.\end{array}$$ Summing the two equations yields that $2x=4+1/4=17/4$, so $x=17/8$. Subtracting the two equations yields that $2y=4-1/4=15/4$, so $y=15/8$. Thus, $xy=\frac{17}{8}\cdot \frac{15}{8}=\boxed{\frac{255}{64}}$.