a) \( \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} \)

To simplify the expression \( \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} \), we'll break down the bases involved: 1. **Rewrite the bases:** - \( 15 = 3 \times 5 \), so \( 15^{n+2} = (3 \times 5)^{n+2} = 3^{n+2} \cdot 5^{n+2} \). - \( 45 = 3^2 \times 5 \), therefore \( 45^{1-n} = (3^2 \times 5)^{1-n} = 3^{2(1-n)} \cdot 5^{1-n} = 3^{2-2n} \cdot 5^{1-n} \). 2. **Combine the terms:** - The expression now becomes: \[ \frac{3^{n+2} \cdot 5^{n+2} \cdot 3^{2-2n} \cdot 5^{1-n}}{3^{3-n}}. \] 3. **Combine the powers of \( 3 \) and \( 5 \):** - For \( 3 \): \[ 3^{n+2 + 2 - 2n} = 3^{n + 4 - 2n} = 3^{4 - n}. \] - For \( 5 \): \[ 5^{n+2 + 1-n} = 5^{n + 2 + 1 - n} = 5^{3}. \] 4. **Rewrite the expression with combined powers:** The simplified expression is: \[ \frac{3^{4-n} \cdot 5^{3}}{3^{3-n}}. \] - Simplifying the powers of \( 3 \): \[ 3^{4-n - (3 - n)} = 3^{4-n - 3 + n} = 3^{1} = 3. \] 5. **Final expression:** Thus, the simplified expression is: \[ 3 \cdot 5^3 = 3 \cdot 125 = 375. \] So the final result is **375**.