a) \( 12 x \cdot 3 x^{2}: x+14 x \cdot x^{3}:\left(1 x^{3}\right) \) b) \( 16 x \cdot x^{3}:(-4)+9 x^{5}: x^{6} \cdot\left(-3 x^{3}\right) \) c) \( 3 x^{2} \cdot\left(10 \cdot 5 x^{3}\right)-10 x^{6} \cdot 6 x^{2}:(2 x) \) d) \( \left(5 x^{2}-2 x^{2}+7 x^{2}\right) \cdot\left(4 x^{3}-x^{3}+6 x^{3}\right) \) e) \( \left(-4 x y^{2}+9 x y^{2}\right):(3 x y+2 x y) \) f) \( \left(x^{3}-8 x^{3}+4 x^{3}\right) \cdot(y-3 y+5 y) \)

First up, let's talk about how algebraic expressions have evolved over time! Ancient civilizations like the Babylonians utilized early forms of algebra to solve complex equations, but it wasn't until the Middle Ages that Islamic mathematicians gave us innovations such as symbols and formalized techniques. Today, what was once doubt and confusion has transformed into a clear, structured vocabulary that makes understanding and manipulating these equations a breeze. Now, if you're tackling these kinds of problems, remember the importance of simplification! Start by looking for common factors and try to factor expressions before performing any operations. This can save you from common mistakes, like forgetting to distribute correctly or losing track of negative signs. It's also a good idea to check your work at each step to catch any errors before they snowball into bigger ones. Happy problem-solving!