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To simplify the expression \( \frac{3^{x+1}-4 \cdot 3^{x-1}}{3^{x-2}} \), we can first rewrite the terms in the numerator. The term \( 3^{x+1} \) can be expressed as \( 3^x \cdot 3^1 = 3 \cdot 3^x \), and \( 4 \cdot 3^{x-1} \) can be rewritten as \( 4 \cdot \frac{3^x}{3} = \frac{4 \cdot 3^x}{3} \). Now the expression looks like this: \[ \frac{3 \cdot 3^x - \frac{4 \cdot 3^x}{3}}{3^{x-2}} \] This simplifies to: \[ \frac{3 \cdot 3^x - \frac{4}{3} \cdot 3^x}{3^{x-2}} = \frac{\left(3 - \frac{4}{3}\right)3^x}{3^{x-2}} \] Combining the terms in the parentheses gives us: \[ \frac{\left(\frac{9}{3} - \frac{4}{3}\right)3^x}{3^{x-2}} = \frac{\frac{5}{3} \cdot 3^x}{3^{x-2}} \] We can simplify further as follows: \[ = \frac{5}{3} \cdot 3^{x - (x-2)} = \frac{5}{3} \cdot 3^2 = \frac{5}{3} \cdot 9 = 15 \] So the simplified expression is: \[ 15 \]
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