1. **State the problem:** Solve the equation $$\frac{3^{z+2}}{9^{z-2}} = 81$$ for $z$. 2. **Rewrite bases to the same base:** Note that $9 = 3^2$ and $81 = 3^4$. So rewrite the equation as: $$\frac{3^{z+2}}{(3^2)^{z-2}} = 3^4$$ 3. **Simplify the denominator exponent:** Using the power of a power rule $\left(a^{m}\right)^n = a^{mn}$: $$(3^2)^{z-2} = 3^{2(z-2)} = 3^{2z - 4}$$ 4. **Rewrite the fraction with the same base:** $$\frac{3^{z+2}}{3^{2z - 4}} = 3^4$$ 5. **Use the quotient rule for exponents:** $$3^{(z+2) - (2z - 4)} = 3^4$$ 6. **Simplify the exponent:** $$(z + 2) - (2z - 4) = z + 2 - 2z + 4 = -z + 6$$ So the equation becomes: $$3^{-z + 6} = 3^4$$ 7. **Since the bases are equal, set the exponents equal:** $$-z + 6 = 4$$ 8. **Solve for $z$:** $$-z = 4 - 6 = -2$$ $$z = 2$$ **Final answer:** $$z = 2$$