3. Simplify. a) \( \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} \) b) \( \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} \) c) \( \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} \) d) \( \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{x^{3}-2 x^{2}}{x^{2}-16} \times \frac{1}{x+4} \) e) \( \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \) f) \( \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \) g) \( \frac{x^{2}+2 x+4}{x^{2} y^{2}-1} \div \frac{x^{3}-8}{x^{2} y+x} \times \frac{x y^{2}-y}{x y} \) h) \( \frac{-2(a-1)(a+2)(a-2)}{2(1-a)} \)

Simplify the expression by following steps: - step0: Simplify: \(\frac{x^{2}+x-6}{3x^{2}-12x}\div \frac{x^{3}-2x^{2}}{x^{2}-16}\times \frac{1}{x+4}\) - step1: Divide the terms: \(\frac{\left(x+3\right)\left(x+4\right)}{3x^{3}}\times \frac{1}{x+4}\) - step2: Reduce the fraction: \(\frac{x+3}{3x^{3}}\times 1\) - step3: Multiply the terms: \(\frac{x+3}{3x^{3}}\) Expand the expression \( \frac{-2(a-1)(a+2)(a-2)}{2(1-a)} \) Simplify the expression by following steps: - step0: Expand the expression: \(\frac{-2\left(a-1\right)\left(a+2\right)\left(a-2\right)}{2\left(1-a\right)}\) - step1: Rewrite the fraction: \(-\frac{2\left(a-1\right)\left(a+2\right)\left(a-2\right)}{2\left(1-a\right)}\) - step2: Reduce the fraction: \(-\frac{\left(a-1\right)\left(a+2\right)\left(a-2\right)}{1-a}\) - step3: Rewrite the expression: \(-\frac{\left(a-1\right)\left(a+2\right)\left(a-2\right)}{-a+1}\) - step4: Calculate: \(\frac{\left(a-1\right)\left(a+2\right)\left(a-2\right)}{a-1}\) - step5: Calculate: \(\frac{a^{3}-a^{2}-4a+4}{a-1}\) - step6: Calculate: \(\frac{\left(a^{2}-4\right)\left(a-1\right)}{a-1}\) - step7: Reduce the fraction: \(a^{2}-4\) Expand the expression \( \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \) Simplify the expression by following steps: - step0: Calculate: \(\frac{a^{2}-2a+1}{a^{2}+2a+1}\times \frac{a^{2}-1}{a^{2}}\times \frac{a^{2}+a}{a^{2}-a}\) - step1: Divide the terms: \(\frac{a^{2}-2a+1}{a^{2}+2a+1}\times \frac{a^{2}-1}{a^{2}}\times \frac{a+1}{a-1}\) - step2: Multiply the terms: \(\frac{\left(a^{2}-2a+1\right)\left(a-1\right)}{a^{2}\left(a+1\right)}\times \frac{a+1}{a-1}\) - step3: Rewrite the expression: \(\frac{\left(a-1\right)^{2}\left(a-1\right)}{a^{2}\left(a+1\right)}\times \frac{a+1}{a-1}\) - step4: Reduce the fraction: \(\frac{\left(a-1\right)\left(a-1\right)}{a^{2}}\times 1\) - step5: Multiply the terms: \(\frac{\left(a-1\right)^{2}}{a^{2}}\) - step6: Calculate: \(\frac{a^{2}-2a+1}{a^{2}}\) Expand the expression \( \frac{x^{2}+2 x+4}{x^{2} y^{2}-1} \div \frac{x^{3}-8}{x^{2} y+x} \times \frac{x y^{2}-y}{x y} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{x^{2}+2x+4}{x^{2}y^{2}-1}\div \frac{x^{3}-8}{x^{2}y+x}\times \frac{xy^{2}-y}{xy}\) - step1: Divide the terms: \(\frac{x^{2}+2x+4}{x^{2}y^{2}-1}\div \frac{x^{3}-8}{x^{2}y+x}\times \frac{xy-1}{x}\) - step2: Divide the terms: \(\frac{x}{\left(xy-1\right)\left(x-2\right)}\times \frac{xy-1}{x}\) - step3: Reduce the fraction: \(\frac{1}{x-2}\times 1\) - step4: Multiply the terms: \(\frac{1}{x-2}\) Expand the expression \( \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{a^{2}b-ab}{a^{3}+a^{2}}\div \frac{a^{2}b-2ab+b}{a^{3}-a}\) - step1: Divide the terms: \(\frac{ab-b}{a^{2}+a}\div \frac{a^{2}b-2ab+b}{a^{3}-a}\) - step2: Divide the terms: \(\frac{ab-b}{a^{2}+a}\div \frac{ab-b}{a^{2}+a}\) - step3: Multiply by the reciprocal: \(\frac{ab-b}{a^{2}+a}\times \frac{a^{2}+a}{ab-b}\) - step4: Reduce the fraction: \(1\times 1\) - step5: Multiply the terms: \(1\) Expand the expression \( \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} \) Simplify the expression by following steps: - step0: Calculate: \(\frac{3x^{2}+27}{2x+6}\times \frac{6x-18}{x^{4}-81}\) - step1: Divide the terms: \(\frac{3x^{2}+27}{2x+6}\times \frac{6}{x^{3}+9x+3x^{2}+27}\) - step2: Rewrite the expression: \(\frac{3\left(x^{2}+9\right)}{2x+6}\times \frac{6}{\left(x^{2}+9\right)\left(x+3\right)}\) - step3: Reduce the fraction: \(\frac{3}{2x+6}\times \frac{6}{x+3}\) - step4: Rewrite the expression: \(\frac{3}{2\left(x+3\right)}\times \frac{6}{x+3}\) - step5: Reduce the fraction: \(\frac{3}{x+3}\times \frac{3}{x+3}\) - step6: Multiply the terms: \(\frac{3\times 3}{\left(x+3\right)\left(x+3\right)}\) - step7: Multiply the terms: \(\frac{9}{\left(x+3\right)\left(x+3\right)}\) - step8: Multiply the terms: \(\frac{9}{\left(x+3\right)^{2}}\) - step9: Calculate: \(\frac{9}{x^{2}+6x+9}\) Expand the expression \( \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{ab-a^{2}}{b^{2}-a^{2}}\div \frac{a^{2}}{b^{2}+ab}\) - step1: Divide the terms: \(\frac{a}{b+a}\div \frac{a^{2}}{b^{2}+ab}\) - step2: Multiply by the reciprocal: \(\frac{a}{b+a}\times \frac{b^{2}+ab}{a^{2}}\) - step3: Reduce the fraction: \(\frac{1}{b+a}\times \frac{b^{2}+ab}{a}\) - step4: Rewrite the expression: \(\frac{1}{b+a}\times \frac{b\left(b+a\right)}{a}\) - step5: Reduce the fraction: \(1\times \frac{b}{a}\) - step6: Multiply the terms: \(\frac{b}{a}\) Expand the expression \( \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{2x^{2}-6x}{x^{2}-1}\div \frac{x-3}{x^{2}-x}\div \frac{1}{3x+3}\) - step1: Divide the terms: \(\frac{2x^{2}}{x+1}\div \frac{1}{3x+3}\) - step2: Multiply by the reciprocal: \(\frac{2x^{2}}{x+1}\times \left(3x+3\right)\) - step3: Rewrite the expression: \(\frac{2x^{2}}{x+1}\times 3\left(x+1\right)\) - step4: Reduce the fraction: \(2x^{2}\times 3\) - step5: Multiply the terms: \(6x^{2}\) Let's simplify each expression step by step: ### a) \[ \frac{a b-a^{2}}{b^{2}-a^{2}} \div \frac{a^{2}}{b^{2}+a b} \] 1. Rewrite the division as multiplication by the reciprocal: \[ \frac{a b-a^{2}}{b^{2}-a^{2}} \times \frac{b^{2}+a b}{a^{2}} \] 2. Factor the numerator and denominator: \[ \frac{a(b-a)}{(b-a)(b+a)} \times \frac{b(b+a)}{a^{2}} \] 3. Cancel common factors: \[ \frac{b}{a} \] ### b) \[ \frac{3 x^{2}+27}{2 x+6} \times \frac{6 x-18}{x^{4}-81} \] 1. Factor each part: \[ \frac{3(x^{2}+9)}{2(x+3)} \times \frac{6(x-3)}{(x^{2}-9)(x^{2}+9)} \] 2. Recognize that \(x^{2}-9\) can be factored as \((x-3)(x+3)\): \[ \frac{3(x^{2}+9)}{2(x+3)} \times \frac{6(x-3)}{(x-3)(x+3)(x^{2}+9)} \] 3. Cancel common factors: \[ \frac{9}{x^{2}+6x+9} \] ### c) \[ \frac{2 x^{2}-6 x}{x^{2}-1} \div \frac{x-3}{x^{2}-x} \div \frac{1}{3 x+3} \] 1. Rewrite the division as multiplication by the reciprocal: \[ \frac{2x(x-3)}{(x-1)(x+1)} \times \frac{x^{2}-x}{x-3} \times \frac{1}{3(x+1)} \] 2. Cancel common factors: \[ 6x^{2} \] ### d) \[ \frac{x^{2}+x-6}{3 x^{2}-12 x} \div \frac{x^{3}-2 x^{2}}{x^{2}-16} \times \frac{1}{x+4} \] 1. Factor each part: \[ \frac{(x-2)(x+3)}{3x(x-4)} \div \frac{x^{2}(x-2)}{(x-4)(x+4)} \times \frac{1}{x+4} \] 2. Rewrite the division as multiplication by the reciprocal: \[ \frac{(x-2)(x+3)}{3x(x-4)} \times \frac{(x-4)(x+4)}{x^{2}(x-2)} \times \frac{1}{x+4} \] 3. Cancel common factors: \[ \frac{x+3}{3x^{3}} \] ### e) \[ \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \] 1. Factor each part: \[ \frac{ab(a-b)}{a^{2}(a+1)} \div \frac{b(a(b-2))}{a(a^{2}-1)} \] 2. Rewrite the division as multiplication by the reciprocal: \[ \frac{ab(a-b)}{a^{2}(a+1)} \times \frac{a(a^{2}-1)}{b(a(b-2))} \] 3. Cancel common factors: \[ 1 \] ### f) \[ \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \] 1. Factor each part: \[ \frac{(a-1)^{2}}{(a+1)^{2}} \times \frac{(a-1)(a+1)}{a^{2}} \times \frac{a(a+1)}{a(a-1)} \] 2. Cancel common factors: \[ \frac{a^{2}-2a+1}{a^{2}} \] ### g) \[ \frac{x^{2}+2 x+4}{x^{2} y^{2}-1} \div \frac{x^{3}-8}{x^{2} y+x} \times \frac{x y^{2}-y}{x y} \] 1. Factor each part: \[ \frac{x^{2}+2x+4}{(xy-1)(xy+1)} \div \frac{(x-2)(x^{2}+2x+4)}{x(y(x+1))} \times \frac{y(x-1)}{xy} \] 2. Rewrite the division as multiplication by the reciprocal: \[ \frac{x^{2}+2x+4}{(xy-1)(xy+1)} \times \frac{y(x(y+1))}{(x-2)(x^{2}+2x+4)} \] 3. Cancel common factors: \[ \frac{1}{x-2} \] ### h) \[ \frac{-2(a-1)(a+2)(a-2)}{2(1-a)} \] 1. Factor out common terms: \[ \frac{-2(a-1)(a+2)(a-2)}{-2(a-1)} \] 2. Cancel common factors: \[ a^{2}-4 \] ### Summary of Results: - a) \( \frac{b}{a} \) - b) \( \frac{9}{x^{2}+6x+9} \) - c) \( 6x^{2} \) - d) \( \frac{x+3}{3x^{3}} \) - e) \( 1 \) - f) \( \frac{a^{2}-2a+1}{a^{2}} \) - g) \( \frac{1}{x-2} \) - h) \( a^{2}-4 \)