Simplify \( \frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}} \)

To simplify the expression \( \frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}} \), we can start by breaking down the numerator. 1. Rewrite the terms in the numerator: \[ 3^{x+1} = 3 \cdot 3^x \] \[ 4 \cdot 3^{x-1} = 4 \cdot \frac{3^x}{3} = \frac{4 \cdot 3^x}{3} \] 2. Substitute these back into the expression: \[ 3^{x+1} - 4 \cdot 3^{x-1} = 3 \cdot 3^x - \frac{4 \cdot 3^x}{3} \] 3. Factor out \( 3^x \) from the numerator: \[ = 3^x \left( 3 - \frac{4}{3} \right) \] To simplify \( 3 - \frac{4}{3} \), we can rewrite \( 3 \) as \( \frac{9}{3} \): \[ = 3^x \left( \frac{9 - 4}{3} \right) = 3^x \left( \frac{5}{3} \right) \] 4. Now plug this back into the entire expression: \[ \frac{3^{x+1} - 4 \cdot 3^{x-1}}{3^{x-2}} = \frac{3^x \cdot \frac{5}{3}}{3^{x-2}} \] 5. Simplifying the denominator, we rewrite \( 3^{x-2} \) as \( \frac{3^x}{3^2} = \frac{3^x}{9} \): \[ = \frac{3^x \cdot \frac{5}{3}}{\frac{3^x}{9}} = \frac{3^x \cdot \frac{5}{3} \cdot 9}{3^x} \] 6. The \( 3^x \) terms cancel out: \[ = \frac{5 \cdot 9}{3} = \frac{45}{3} = 15 \] Thus, the simplified expression is \[ \boxed{15} \]