Exercise 2.5 1. Simplify without using a calculator. \( \begin{array}{ll}\text { a) } \frac{20^{3} \times 25^{2} \times 4}{10^{7}} & \text { b) } \frac{10^{3} \cdot 15^{4} \cdot 6^{2}}{20^{4} \cdot 12.45^{2}} \\ \text { c) } \frac{2 \times 2^{2 a} \times 16^{2}}{4^{a+5}} & \text { d) } \frac{3^{2} \times 3^{n-1} \times 3^{0}}{9^{n-1}} \\ \text { e) } \frac{2^{m+n}}{2^{n}} & \text { f) } \frac{a^{a+b+c}}{a^{2 a-2 b+c}} \\ \text { g) } 4^{x} \times 8^{x+1} & \text { h) } \frac{9^{x}}{27^{x-1}} \\ \text { i) } \frac{5^{2 n+1} \times 3^{2 n-3}}{3^{2 n} \times 5^{2 n}} & \text { j) } \frac{5^{2 n+1} \times 5^{2 n-2} \times 5}{25^{2 n-1}} \\ \text { k) } \frac{12^{n+1} \times 9^{2 n-1}}{36^{n} \times 8^{1-n}} & \text { l) } \frac{5^{2 n} \times 15^{n-1} \times 3^{n}}{125^{n} \times 3^{n-1}} \\ \text { m) } \frac{27^{n-2} \times 6^{n}}{162^{n}} & \text { n) } \frac{10^{n} \times 25^{n-1} \times 2}{50^{n+1}} \\ \text { o) } \frac{12^{x} \times 9^{x+1}}{4^{x-1} \times 27^{x}} & \text { p) } \frac{20^{x+1} \times 4^{x}}{16^{x-1} \times 5^{x}} \\ \text { 2. Multiply. } & \text { b) } y\left(y^{-1}-2\right) \\ \text { a) } x^{2}\left(x^{3}+1\right) & \text { d) } 3^{x+1}\left(3^{x}-3^{2}\right) \\ \text { c) } 2^{n}\left(2^{n}+1\right) & \\ \text { Eind } & \text { it }\end{array} \)
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To solve these expressions, a strong foundational understanding of exponents and how to manipulate them is essential. Remember, when you multiply like bases, you add the exponents, and when you divide, you subtract them. This can transform complex-looking fractions into much simpler forms. For example, in expression (a), simplify \(20\) and \(25\) into their prime factorizations to reduce the expression effectively. In terms of practical application, simplifying expressions with exponents and multiplication is incredibly useful in fields like engineering and finance, where you often need to analyze growth rates, interest calculations, and more without getting bogged down by cumbersome calculations. Seamless simplifications save time and minimize errors, allowing you to focus on interpreting the results accurately.