If $3^{x + y} = 81$ and $81^{x - y} = 3,$ then what is the value of the product $xy$? Express your answer as a common fraction.
Since $81 = 3^4$, then $3 = 81^{1/4}$. Comparing exponents, it follows that we have the system of equations \begin{align*}
x+y &= 4 \\
x -y &= 1/4.
\end{align*} 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, $xy = \frac{17}{8} \cdot \frac{15}{8} = \boxed{\frac{255}{64}}$.