Section 11-1: Properties of Exponents Property of Negatives: Suppose m and n are positive integers and a and b are real numbers. Then the following are true: Property Definition Example    Property of 1:  Property of 0:  Property of Negatives: Product Property  Power of a Power  Power of a Quotient  Power of a Product  Quotient Property 𝑏 𝑛 = b 5 1 = 5 𝑏 0 = 1 7 0 = 1 𝑏 −𝑛 = 1 𝑏 𝑛 4 −2 = 1 4 2 = 1 16 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚+𝑛 3 2 3 4 = 3 2+4 = 3 6 𝑎 𝑚 𝑛 = 𝑎 𝑚𝑛 2 𝑥 2 𝑦 3 = 8 𝑥 6 𝑦 3 𝑎 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 𝑥 3 𝑦 2 4 = 𝑥 4 81 𝑦 8 𝑎𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 3𝑥𝑦 3 2 = 9 𝑥 2 𝑦 6 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 𝑥 5 𝑥 2 = 𝑥 3 or 𝑥 3 𝑥 9 = 1 𝑥 6

c. a. b. d. 5-4 e. f. Example 1 Evaluate each expression. 4 5 ∙ 4 3 4 4 = 4 8 4 4 = 4 4 = 256 = 1 7 8 = 8 7 a. b. = 1 5 4 = 1 625 c. = 3 4 1 4 = 3 d. 5-4 e. f. = 3 5 3 1 = 3 6 = 729 (3a-2)3•3a5 = 27 𝑎 −6 3 𝑎 5 = 81 𝑎 −1 = 81 𝑎

Example 3 Simplify each expression. 𝑥 2 𝑦 5 𝑦 3 5 a. b. (s5t2)3 = 𝑠 15 𝑡 6 = 𝑥 2 𝑦 5 𝑦 15 = 𝑥 2 𝑦 10 Example 3 Simplify each expression. =16 1 4 𝑎 2 = 2 4 1 4 𝑎 2 = 2𝑎 2 a. b. = 16 1 6 𝑥 2 6 = 2 4 1 6 𝑥 1 3 = 2 2 3 𝑥 1 3 = 3 4𝑥

Example 4 Evaluate each expression. = 3 5 3 5 = 3 3 = 27 a. b. = 27 2 3 − 1 3 = 27 1 3 = 3 3 1 3 = 3 Example 5 a. Express using rational exponents.   b. Express using a radical. = 32 1 5 𝑠 25 5 𝑡 10 5 = 2 5 1 5 𝑠 5 𝑡 2 = 2𝑠 5 𝑡 2 20 4 𝑥 3 𝑦

𝑟 5 𝑠 15 𝑡 4 Example 6 Simplify: = 𝑟 2 𝑠 7 𝑡 2 𝑟𝑠 = 𝑟 2 𝑠 7 𝑡 2 𝑟𝑠 Example 7 Solve: 333 = - 10.   343= 𝑥 3 2 343 2 3 = 𝑥 3 2 2 3 7 3 2 3 = x 7 2 = x 49 = x