9. Simplify: \( \frac{9 \times 3^{n+1}-3^{n+2}}{3^{n+1}-3^{n}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(9\times 3^{n+1}-3^{n+2}\right)}{\left(3^{n+1}-3^{n}\right)}\) - step1: Remove the parentheses: \(\frac{9\times 3^{n+1}-3^{n+2}}{3^{n+1}-3^{n}}\) - step2: Multiply the terms: \(\frac{3^{n+3}-3^{n+2}}{3^{n+1}-3^{n}}\) - step3: Subtract the terms: \(\frac{2\times 3^{n+2}}{3^{n+1}-3^{n}}\) - step4: Subtract the terms: \(\frac{2\times 3^{n+2}}{2\times 3^{n}}\) - step5: Reduce the fraction: \(\frac{3^{n+2}}{3^{n}}\) - step6: Divide the numbers: \(3^{n+2-n}\) - step7: Subtract the terms: \(3^{2}\) - step8: Evaluate the power: \(9\) To simplify the expression \( \frac{9 \times 3^{n+1}-3^{n+2}}{3^{n+1}-3^{n}} \), we can follow these steps: 1. **Factor the numerator**: - The numerator is \( 9 \times 3^{n+1} - 3^{n+2} \). - We can factor out \( 3^{n+1} \): \[ 9 \times 3^{n+1} - 3^{n+2} = 3^{n+1}(9 - 3) = 3^{n+1} \times 6 \] 2. **Factor the denominator**: - The denominator is \( 3^{n+1} - 3^n \). - We can factor out \( 3^n \): \[ 3^{n+1} - 3^n = 3^n(3 - 1) = 3^n \times 2 \] 3. **Rewrite the expression**: - Now we can rewrite the original expression using the factored forms: \[ \frac{3^{n+1} \times 6}{3^n \times 2} \] 4. **Simplify the expression**: - Cancel \( 3^n \) from the numerator and denominator: \[ = \frac{3^{n+1}}{3^n} \times \frac{6}{2} = 3^{1} \times 3 = 3 \times 3 = 9 \] Thus, the simplified result is \( 9 \).