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

To simplify the expression \( \frac{2 \cdot 3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}} \), we first factor out the common terms in the numerator. Both terms in the numerator can be expressed in terms of \( 3^{x-3} \): \[ 2 \cdot 3^{x+2} = 2 \cdot 3^{x-3} \cdot 3^5 = 2 \cdot 3^{x-3} \cdot 243 \] \[ 3^{x-3} = 1 \cdot 3^{x-3} \] Now, combining those gives us: \[ = 3^{x-3} (2 \cdot 243 + 1) = 3^{x-3} (486 + 1) = 3^{x-3} \cdot 487 \] Now, replace the numerator in the original fraction: \[ \frac{3^{x-3} \cdot 487}{5 \cdot 3^{x-2}} \] Since \( 3^{x-3} \) can be divided by \( 3^{x-2} \): \[ \frac{487}{5} \cdot \frac{3^{x-3}}{3^{x-2}} = \frac{487}{5} \cdot \frac{1}{3^{1}} = \frac{487}{15} \] So the simplified form is: \[ \frac{487}{15} \]