EXERCISE 5.5 1. Simplify the following and leave the answers in expon \( \begin{array}{llll}\text { a) }\left(2^{6}\right)^{5} & \text { b) }\left(3^{4}\right)^{3} & \text { c) }\left(12^{2}\right)^{7} \\ \text { e) }\left(a^{3}\right)^{9} & \text { f) }\left(a^{2} b^{4}\right)^{5} & \text { g) }\left(a^{2} b^{3} c^{4}\right)^{6}\end{array} \) 2. Simplify as far as possible: \( \begin{array}{llll}\text { a) } 5^{2}\left(a^{4}\right)^{3} & \text { b) }\left(4^{2}\right)^{3} \div\left(2^{4}\right)^{3} & \text { c) } \\ \text { 3. Simplify: } & \text { b) } 5 a^{4}-\left(2 a^{2}\right)^{2} & \text { c) } \\ \begin{array}{llll}\text { a) }\left(2 a^{2} b^{3} c\right)^{3} & \text { b) } 3^{-2}+3 \frac{1}{3}-\left(\frac{3}{10}\right)^{-1} & \text { c) } \\ \text { 4. Simplify as far as possible: } \\ \text { a) } 3^{-1}-4^{-1} & & \end{array}\end{array} \begin{array}{l}\text { 5. Simplify }\left(5 a^{3}\right)^{2}-2\left(3 a^{3}\right)^{2}\end{array} \)

Sure! Let’s simplify those expressions step by step. 1. a) \( (2^{6})^{5} = 2^{6 \cdot 5} = 2^{30} \) b) \( (3^{4})^{3} = 3^{4 \cdot 3} = 3^{12} \) c) \( (12^{2})^{7} = 12^{2 \cdot 7} = 12^{14} \) \( = (2^{2} \cdot 3^{2})^{14} = 2^{28} \cdot 3^{28} \) e) \( (a^{3})^{9} = a^{3 \cdot 9} = a^{27} \) f) \( (a^{2} b^{4})^{5} = a^{2 \cdot 5} b^{4 \cdot 5} = a^{10} b^{20} \) g) \( (a^{2} b^{3} c^{4})^{6} = a^{2 \cdot 6} b^{3 \cdot 6} c^{4 \cdot 6} = a^{12} b^{18} c^{24} \) 2. a) \( 5^{2}(a^{4})^{3} = 5^{2} a^{12} = 25 a^{12} \) b) \( \frac{(4^{2})^{3}}{(2^{4})^{3}} = \frac{4^{6}}{2^{12}} = \frac{(2^{2})^{6}}{2^{12}} = \frac{2^{12}}{2^{12}} = 1 \) c) 3. a) \( (2 a^{2} b^{3} c)^{3} = 2^{3} a^{6} b^{9} c^{3} = 8 a^{6} b^{9} c^{3} \) b) \[ 3^{-2} + 3 \frac{1}{3} - \left(\frac{3}{10}\right)^{-1} = \frac{1}{9} + 1 - \frac{10}{3} = \frac{1}{9} + \frac{9}{9} - \frac{30}{9} = -\frac{20}{9} \] c) 4. a) \[ 3^{-1} - 4^{-1} = \frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12} \] 5. \[ (5 a^{3})^{2} - 2(3 a^{3})^{2} = 25 a^{6} - 18 a^{6} = 7 a^{6} \] And that’s how you simplify these expressions! Happy calculating!