Chapter 5: Exponents and Logarithms Section 5.1: Growth and Decay: Integral Exponents

Laws of Exponents Same bases: 1)b x · b y = b x + y 2) b x = b x – y (b ≠ 0) byby 3) If b ≠ 0, 1, or -1, then b x = b y if and only if x = y.

Same Exponents: 4)(ab) x = a x b x 5) a x = a x (b ≠0) b b x 6) If x ≠ 0, a > 0, and b > 0, then a x = b x if and only if a = b.

Power of a power: 7) (b x ) y = b xy

Definition of b 0 : If law 1 is to hold for y = 0, then we have b x · b 0 = b x + 0 = b x Since b 0 behaves like the number 1, we define it to be 1: b 0 = 1 (b ≠ 0)

Definition of b -x : If law 1 is to hold for y = -x, and b ≠ 0, then we have b x · b -x = b x + (-x) = b 0 = 1 Since b x and b -x have a product of 1, they are reciprocals of each other. Therefore, we define: b -x = 1 (x > 0 and b ≠ 0) b x

We shall assume throughout the rest of this book that variables are restricted so that there are no denominators of zero. In this section only, we assume that variables appearing as exponents represent integers.

Example 1: Simplify. a)8 -2 = = 1 64 b)4 · 3 -2 = 4 · 1/9 = 4/9

c) 3a 3 · 6a 6 a -1 = 3a 3 · 6a 6 · a 1 = 18a 10 d) 3a 3 - 6a 6 a -1 = 3a 3 - 6a 6 a -1 a -1 = 3a 4 – 6a 7

Example 2: Simplify each expression. a)(4 · 3) -2 = = b) (2 -1 · 4 -1 ) -1 = 2 · 4 = 8

HOMEWORK p. 173; 2 – 12 even (parts a and b), 22, 24 (do not complete #8b or #12a!)