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

To simplify the expression \( \frac{2 \cdot 3^{x+3}+3^{x-3}}{5 \cdot 3^{x-2}} \), we can start by rewriting both terms in the numerator to have a common base. The numerator can be factored as follows: \[ 2 \cdot 3^{x+3} + 3^{x-3} = 2 \cdot 3^{x+3} + 3^{x-3} \] Notice that \( 3^{x+3} = 3^{x-2} \cdot 3^5 \) and \( 3^{x-3} = 3^{x-2} \cdot 3^{-1} \). Thus, we can rewrite the numerator: \[ = 2 \cdot 3^{x-2} \cdot 3^5 + 3^{x-2} \cdot 3^{-1} \] Factoring out \( 3^{x-2} \): \[ = 3^{x-2} \left( 2 \cdot 3^5 + 3^{-1} \right) \] Now we can substitute this back into our original expression: \[ \frac{3^{x-2} \left( 2 \cdot 3^5 + 3^{-1} \right)}{5 \cdot 3^{x-2}} \] The \( 3^{x-2} \) cancels out: \[ = \frac{2 \cdot 3^5 + 3^{-1}}{5} \] Finally, substituting \( 3^5 = 243 \) and \( 3^{-1} = \frac{1}{3} \): \[ = \frac{2 \cdot 243 + \frac{1}{3}}{5} = \frac{486 + \frac{1}{3}}{5} = \frac{\frac{1458 + 1}{3}}{5} = \frac{1459}{15} \] So, the simplified expression is: \[ \frac{1459}{15} \]