1. **State the problem:** Solve the equation $$9^{3x - 7} = 81^{x + 2}$$. 2. **Rewrite bases as powers of the same base:** Since $$9 = 3^2$$ and $$81 = 3^4$$, rewrite the equation as: $$ (3^2)^{3x - 7} = (3^4)^{x + 2} $$ 3. **Apply power of a power rule:** $$ 3^{2(3x - 7)} = 3^{4(x + 2)} $$ 4. **Simplify exponents:** $$ 3^{6x - 14} = 3^{4x + 8} $$ 5. **Since the bases are equal, set exponents equal:** $$ 6x - 14 = 4x + 8 $$ 6. **Solve for $$x$$:** $$ 6x - 4x = 8 + 14 $$ $$ 2x = 22 $$ $$ x = \frac{22}{2} $$ $$ x = 11 $$ 7. **Final answer:** The solution set is $$\{11\}$$.