1. **State the problem:** Given the equation $9^x = 81^y$, find the value of $\frac{9}{x}$. 2. **Rewrite the bases as powers of 3:** Since $9 = 3^2$ and $81 = 3^4$, rewrite the equation as $$ (3^2)^x = (3^4)^y $$ 3. **Apply the power of a power rule:** $$ 3^{2x} = 3^{4y} $$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$ 2x = 4y $$ 5. **Solve for $x$ in terms of $y$:** $$ x = \frac{4y}{2} = 2y $$ 6. **Find $\frac{9}{x}$:** Substitute $x = 2y$: $$ \frac{9}{x} = \frac{9}{2y} $$ **Final answer:** $$ \boxed{\frac{9}{2y}} $$