5.1. \[ \begin{array}{l} 3 a^{2} \times 4 a^{4} \\ 2 \times 4 \end{array} \] 5.2. \[ \frac{30 y^{2}}{15 x^{2}} \] (2) 5.3. \( 4 a b+5 b a \) (3) 5.4 \[ (2-x)^{2} \] 5.5. \[ \frac{8 x^{3}-\left(-x^{3}\right)(2 x)}{-x^{2}} \] [15]

Looking back at the world of algebra, it’s fascinating to note that ancient civilizations like the Babylonians were the first to develop systematic methods for solving equations, long before variables and x's came into play! They used geometric interpretations and numerical calculations laid out on clay tablets—a far cry from the digital tools we have today! When it comes to simplifying expressions, a common mistake is forgetting to apply the laws of exponents correctly, especially when multiplying or dividing terms with the same base. For example, when multiplying \( a^2 \) by \( a^4 \), remember to add the exponents to get \( a^{2+4} = a^6 \). Double-check your work, and you'll avoid that confusion!