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) \( \frac{x^{2}+2 x+4}{x^{2} y^{2}-1} \div \frac{x^{3}-8}{x^{2} y+x} \times \frac{x y^{2}-y}{x y} \)

When you simplify these expressions, keep in mind that factoring is your best friend! For the first problem, factor out common terms in the numerators and denominators before you begin dividing or multiplying. This will help you cancel out terms quickly and efficiently. Always check your factorizations twice to avoid missing a potential simplification! In real-world applications, the ability to simplify complex algebraic expressions is key in fields like engineering and physics, where you might be working with equations that model real systems. Simplification allows for easier computations and helps in deriving practical conclusions from complex mathematical models. Whether you're calculating loads in structures or analyzing electrical circuits, being adept at simplification will save time and minimize errors!