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

Simplify the expression by following steps: - step0: Solution: \(2\times 3^{x}+3^{x-2}\) The expression \( 2 \cdot 3^{x} + 3^{x-2} \) can be simplified, but it remains in the same form as \( 2 \times 3^{x} + 3^{x-2} \). To further analyze or manipulate this expression, we can factor it or combine like terms if possible. Let's rewrite \( 3^{x-2} \) in terms of \( 3^{x} \): \[ 3^{x-2} = \frac{3^{x}}{3^{2}} = \frac{3^{x}}{9} \] Now, substituting this back into the expression gives: \[ 2 \cdot 3^{x} + \frac{3^{x}}{9} \] Next, we can factor out \( 3^{x} \): \[ 3^{x} \left( 2 + \frac{1}{9} \right) \] Now, let's simplify \( 2 + \frac{1}{9} \): \[ 2 = \frac{18}{9} \quad \text{so} \quad 2 + \frac{1}{9} = \frac{18}{9} + \frac{1}{9} = \frac{19}{9} \] Thus, the expression can be rewritten as: \[ 3^{x} \cdot \frac{19}{9} \] So, the final simplified form of the expression \( 2 \cdot 3^{x} + 3^{x-2} \) is: \[ \frac{19}{9} \cdot 3^{x} \]