1. Simplify: \( \begin{array}{ll}\text { (a) } \frac{3 a^{2}}{4 a^{3}} & \text { (b) } \frac{5 x^{2} y}{10 x y^{2}} \\ \begin{array}{ll}\text { (d) } \frac{5 a^{3}-45 a}{4 a^{2}-12 a} & \text { (e) } \frac{(x-3)^{3}}{2 x-6} \\ \begin{array}{ll}\text { (g) } \frac{a^{2}+6 x+8}{a^{2}-16} & \text { (h) } \frac{x^{2}+x-12}{x^{2}-x-6} \\ \begin{array}{ll}x^{2}+9 x+20 & \text { (k) } \frac{x^{2}+5 x+6}{(x+3)^{2}} \\ \text { (m) } \frac{x^{2}-1}{x^{2}-6 x+5} & \text { (n) } \frac{3 x y z}{3 x^{2}-12 x}\end{array}\end{array} ⿴ \begin{array}{ll}\text { (f) }\end{array}\end{array} .\end{array} \)

Simplifying fractions is a fantastic way to get to the essence of algebra! Back in ancient Greece, mathematicians like Euclid were already exploring similar problems, paving the way for the art of simplification we use today. Just think, every time you reduce a fraction, you're doing math that has roots in over two thousand years of intellectual history! Now, let’s turn our attention to real-world applications. Simplifying fractions is not just a math class exercise; it has practical uses in cooking, construction, and even finance! Whether you're adjusting a recipe for a smaller group or calculating the best deal on a bulk purchase, understanding how to simplify can save you both time and resources. So, next time you handle a fraction, remember, you’re sharpening a skill that has real benefits in daily life!