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

Did you know that simplifying algebraic expressions like these has roots that go all the way back to ancient civilizations? The Babylonians were among the first to use algebraic ideas around 2000 BC, laying the groundwork for the symbols and structures we use today. They wouldn’t have seen the variables we recognize now, but their calculations and problem-solving techniques were impressively sophisticated for their time! When tackling complex expressions, it's crucial to watch for common mistakes, like overlooking common factors during simplification. For instance, when you factor both the numerator and denominator, ensure you simplify correctly and check for any restrictions on variable values. Remember, algebra can sometimes be as tricky as a game of chess – one wrong move can derail your whole strategy!