(2) \( \frac{5^{x} \cdot 9^{x+1}}{3^{x+2} \cdot 15^{x-1}} \)

To simplify the expression \( \frac{5^{x} \cdot 9^{x+1}}{3^{x+2} \cdot 15^{x-1}} \), start by expressing all terms with common bases: 1. Rewrite \( 9 \) as \( 3^2 \) and \( 15 \) as \( 3 \cdot 5 \): \[ 9^{x+1} = (3^2)^{x+1} = 3^{2(x+1)} = 3^{2x + 2} \] \[ 15^{x-1} = (3 \cdot 5)^{x-1} = 3^{x-1} \cdot 5^{x-1} \] 2. Substitute these back into the original expression: \[ \frac{5^{x} \cdot 3^{2x + 2}}{3^{x+2} \cdot (3^{x-1} \cdot 5^{x-1})} \] 3. This simplifies to: \[ \frac{5^{x} \cdot 3^{2x + 2}}{3^{x + 2 + x - 1} \cdot 5^{x-1}} = \frac{5^{x} \cdot 3^{2x + 2}}{3^{2x + 1} \cdot 5^{x - 1}} \] 4. Now, separate the bases: \[ = \frac{5^{x}}{5^{x - 1}} \cdot \frac{3^{2x + 2}}{3^{2x + 1}} = 5^{x - (x - 1)} \cdot 3^{(2x + 2) - (2x + 1)} \] 5. This results in: \[ = 5^{1} \cdot 3^{1} = 5 \cdot 3 = 15 \] Thus, the simplified expression is \( 15 \).