1. **State the problem:** Simplify the expression $$\frac{3^{n+2}}{3^{n+3} - 3^{n+1}}$$. 2. **Recall the properties of exponents:** - $$a^{m} \times a^{n} = a^{m+n}$$ - $$\frac{a^{m}}{a^{n}} = a^{m-n}$$ - Factorization can help simplify expressions. 3. **Rewrite the denominator:** $$3^{n+3} - 3^{n+1} = 3^{n+1} \times (3^{2} - 1) = 3^{n+1} \times (9 - 1) = 3^{n+1} \times 8$$ 4. **Substitute back into the expression:** $$\frac{3^{n+2}}{3^{n+1} \times 8}$$ 5. **Simplify the fraction by canceling common factors:** $$\frac{3^{n+2}}{3^{n+1} \times 8} = \frac{3^{\cancel{n+1}+1}}{3^{\cancel{n+1}} \times 8} = \frac{3^{1}}{8} = \frac{3}{8}$$ 6. **Final answer:** $$\boxed{\frac{3}{8}}$$