e) \( \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \) f) \( \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(a^{2}b-ab\right)}{\left(a^{3}+a^{2}\right)}}{\left(\frac{\left(a^{2}b-2ab+b\right)}{\left(a^{3}-a\right)}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{a^{2}b-ab}{a^{3}+a^{2}}}{\frac{a^{2}b-2ab+b}{a^{3}-a}}\) - step2: Divide the terms: \(\frac{\frac{ab-b}{a^{2}+a}}{\frac{a^{2}b-2ab+b}{a^{3}-a}}\) - step3: Divide the terms: \(\frac{\frac{ab-b}{a^{2}+a}}{\frac{ab-b}{a^{2}+a}}\) - step4: Divide the terms: \(1\) Calculate or simplify the expression \( (a^2 - 2*a + 1)/(a^2 + 2*a + 1) * (a^2 - 1)/(a^2) * (a^2 + a)/(a^2 - a) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\frac{\left(a^{2}-2a+1\right)}{\left(a^{2}+2a+1\right)}\times \left(a^{2}-1\right)}{a^{2}}\times \left(a^{2}+a\right)}{\left(a^{2}-a\right)}\) - step1: Remove the parentheses: \(\frac{\frac{\frac{a^{2}-2a+1}{a^{2}+2a+1}\times \left(a^{2}-1\right)}{a^{2}}\times \left(a^{2}+a\right)}{a^{2}-a}\) - step2: Multiply the terms: \(\frac{\frac{\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a+1}}{a^{2}}\times \left(a^{2}+a\right)}{a^{2}-a}\) - step3: Divide the terms: \(\frac{\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a^{2}\left(a+1\right)}\times \left(a^{2}+a\right)}{a^{2}-a}\) - step4: Multiply the terms: \(\frac{\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a}}{a^{2}-a}\) - step5: Multiply by the reciprocal: \(\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a}\times \frac{1}{a^{2}-a}\) - step6: Rewrite the expression: \(\frac{\left(a-1\right)^{2}\left(a-1\right)}{a}\times \frac{1}{a\left(a-1\right)}\) - step7: Reduce the fraction: \(\frac{\left(a-1\right)\left(a-1\right)}{a}\times \frac{1}{a}\) - step8: Multiply the terms: \(\frac{\left(a-1\right)^{2}}{a\times a}\) - step9: Multiply the terms: \(\frac{\left(a-1\right)^{2}}{a^{2}}\) - step10: Calculate: \(\frac{a^{2}-2a+1}{a^{2}}\) Let's break down the solutions for both parts step by step. ### Part e We need to simplify the expression: \[ \frac{a^{2} b - a b}{a^{3} + a^{2}} \div \frac{a^{2} b - 2 a b + b}{a^{3} - a} \] 1. **Simplify the first fraction:** - The numerator: \( a^{2} b - a b = a b (a - 1) \) - The denominator: \( a^{3} + a^{2} = a^{2}(a + 1) \) - So, the first fraction becomes: \[ \frac{a b (a - 1)}{a^{2}(a + 1)} \] 2. **Simplify the second fraction:** - The numerator: \( a^{2} b - 2 a b + b = b(a^{2} - 2a + 1) = b(a - 1)^{2} \) - The denominator: \( a^{3} - a = a(a^{2} - 1) = a(a - 1)(a + 1) \) - So, the second fraction becomes: \[ \frac{b(a - 1)^{2}}{a(a - 1)(a + 1)} \] 3. **Dividing the two fractions:** - Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{a b (a - 1)}{a^{2}(a + 1)} \times \frac{a(a - 1)(a + 1)}{b(a - 1)^{2}} \] 4. **Canceling common terms:** - The \( b \) cancels out. - One \( (a - 1) \) cancels out. - The \( a \) in the numerator and one \( a \) in the denominator cancels out. - This simplifies to: \[ 1 \] ### Part f We need to simplify the expression: \[ \frac{a^{2} - 2 a + 1}{a^{2} + 2 a + 1} \times \frac{a^{2} - 1}{a^{2}} \times \frac{a^{2} + a}{a^{2} - a} \] 1. **Simplify each fraction:** - The first fraction: \[ \frac{(a - 1)^{2}}{(a + 1)^{2}} \] - The second fraction: \[ \frac{(a - 1)(a + 1)}{a^{2}} \] - The third fraction: \[ \frac{a(a + 1)}{a(a - 1)} = \frac{a + 1}{a - 1} \] 2. **Combine the fractions:** - The entire expression becomes: \[ \frac{(a - 1)^{2}}{(a + 1)^{2}} \times \frac{(a - 1)(a + 1)}{a^{2}} \times \frac{a + 1}{a - 1} \] 3. **Canceling common terms:** - The \( (a - 1) \) cancels out. - The \( (a + 1) \) cancels out. - This simplifies to: \[ \frac{(a - 1)}{a^{2}} \] ### Final Results - For part e, the simplified result is: \[ 1 \] - For part f, the simplified result is: \[ \frac{a^{2} - 2a + 1}{a^{2}} \]