d.) \( \frac{2 \cdot 3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}} \)

To simplify \( \frac{2 \cdot 3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}} \), we can first rewrite the numerator by factoring out the common term, \( 3^{x-3} \): 1. The numerator becomes: \[ 2 \cdot 3^{x+2} + 3^{x-3} = 2 \cdot 3^{x-3} \cdot 3^5 + 3^{x-3} = 3^{x-3}(2 \cdot 3^5 + 1) \] Here, \( 3^5 \) is calculated as \( 243 \), so we have: \[ 2 \cdot 243 + 1 = 486 + 1 = 487 \] Thus, the numerator simplifies to \( 487 \cdot 3^{x-3} \). 2. The denominator is: \[ 5 \cdot 3^{x-2} \] Now, we can substitute these back into the fraction: \[ \frac{487 \cdot 3^{x-3}}{5 \cdot 3^{x-2}} \] 3. By simplifying: \[ = \frac{487}{5} \cdot \frac{3^{x-3}}{3^{x-2}} = \frac{487}{5} \cdot \frac{1}{3} = \frac{487}{15} \] Thus, the simplified expression is: \[ \frac{487}{15} \]