1. **State the problem:** Solve the equation $$\frac{3^{z+2}}{9^{z-2}} = 81$$ for $z$. 2. **Rewrite the bases:** Note that $9 = 3^2$ and $81 = 3^4$. Substitute these into the equation: $$\frac{3^{z+2}}{(3^2)^{z-2}} = 3^4$$ 3. **Simplify the denominator:** Use the power of a power rule $\left(a^m\right)^n = a^{mn}$: $$\frac{3^{z+2}}{3^{2(z-2)}} = 3^4$$ 4. **Divide powers with the same base:** Use $\frac{a^m}{a^n} = a^{m-n}$: $$3^{z+2 - 2(z-2)} = 3^4$$ 5. **Simplify the exponent:** $$z + 2 - 2z + 4 = 4$$ $$-z + 6 = 4$$ 6. **Solve for $z$:** $$-z = 4 - 6$$ $$-z = -2$$ $$z = 2$$ **Final answer:** $$z = 2$$