1. **State the problem:** Simplify the expression $$3^{n+2} + \left(3^{n+3} - 3^{n+1}\right)$$ and determine which of the given options it equals. 2. **Recall the properties of exponents:** - $$a^{m} \times a^{k} = a^{m+k}$$ - $$a^{m} / a^{k} = a^{m-k}$$ - We can factor expressions with common bases and exponents. 3. **Rewrite the expression:** $$3^{n+2} + 3^{n+3} - 3^{n+1}$$ 4. **Factor out the smallest power of 3, which is $$3^{n+1}$$:** $$3^{n+1} \left(3^{1} + 3^{2} - 1\right) = 3^{n+1} \left(3 + 9 - 1\right)$$ 5. **Simplify inside the parentheses:** $$3 + 9 - 1 = 11$$ 6. **So the expression becomes:** $$3^{n+1} \times 11 = 11 \times 3^{n+1}$$ 7. **Compare with the options:** None of the options match $$11 \times 3^{n+1}$$ or its reciprocal forms. 8. **Check if the problem expects a simplified fraction or reciprocal:** The options are all fractions with powers of 3 in the denominator or constants. 9. **Since the expression simplifies to $$11 \times 3^{n+1}$$, it does not equal any of the given options (A), (B), (C), or (D).** **Final answer:** The expression simplifies to $$11 \times 3^{n+1}$$ and does not match any of the provided options.