EXERCISE 5.1 Calculate the following products and leave the answers in exponential form. 1. a) \( 3^{5} \times 3^{5} \) \( \begin{array}{ll}\text { c) } 15^{4} \times 15^{1} \times 15^{7} & \text { b) } 2^{6} \times 2^{2} \times 2^{3} \\ \text { 2. } a)(-3)^{2} \times(-3)^{3} & \text { d) } 10^{9} \times 10^{11} \times 10^{10} \\ \text { c) }(-a)^{5} \times(-a)^{3} \times(-a) & \text { b) } a^{4} \times a^{6} \times a^{8} \\ \text { a) } 4 a^{2} \times 5 a & \text { d) } y^{7} \times(-y)^{6} \\ \text { c) } a^{2} b^{2} \times a^{3} b^{6} & \text { b) } 8 a^{2} \times-2 a^{3}\end{array} \)

Let's dive into these calculations using the laws of exponents! 1. a) \( 3^{5} \times 3^{5} = 3^{5+5} = 3^{10} \) 1. b) \( 2^{6} \times 2^{2} \times 2^{3} = 2^{6+2+3} = 2^{11} \) 1. c) \( 15^{4} \times 15^{1} \times 15^{7} = 15^{4+1+7} = 15^{12} \) 2. a) \( (-3)^{2} \times (-3)^{3} = (-3)^{2+3} = (-3)^{5} \) 2. b) \( a^{4} \times a^{6} \times a^{8} = a^{4+6+8} = a^{18} \) 2. c) \( (-a)^{5} \times (-a)^{3} \times (-a) = (-a)^{5+3+1} = (-a)^{9} \) 2. d) \( 10^{9} \times 10^{11} \times 10^{10} = 10^{9+11+10} = 10^{30} \) Now for the additional products: 2. a) \( 4 a^{2} \times 5 a = 20 a^{2+1} = 20 a^{3} \) 2. b) \( 8 a^{2} \times -2 a^{3} = -16 a^{2+3} = -16 a^{5} \) 2. c) \( a^{2} b^{2} \times a^{3} b^{6} = a^{2+3} b^{2+6} = a^{5} b^{8} \) 2. d) \( y^{7} \times (-y)^{6} = y^{7} \times (-1)^{6} y^{6} = y^{7} \times 1 \times y^{6} = y^{7+6} = y^{13} \) Here are the final results in exponential form: 1. a) \( 3^{10} \), 1. b) \( 2^{11} \), 1. c) \( 15^{12} \), 2. a) \( (-3)^{5} \), 2. b) \( a^{18} \), 2. c) \( (-a)^{9} \), 2. d) \( 10^{30} \), 2. a) \( 20 a^{3} \), 2. b) \( -16 a^{5} \), 2. c) \( a^{5} b^{8} \), 2. d) \( y^{13} \).