University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

229

Pages:

11
Author:

Keenan Fowler

§ 12.1 Exponents Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3 • 3 • 3 • 3 3 is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations. 4 Evaluating Exponential Expressions Example Evaluate each of the following expressions. 34 = 3 • 3 • 3 • 3 = 81 (–5)2 = (– 5)(–5) = 25 –62 = – (6)(6) = –36 (2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512 3 • 42 = 3 • 4 • 4 = 48 5 Evaluating Exponential Expressions Example Evaluate each of the following expressions. a.) Find 3x2 when x = 5. 3x2 = 3(5)2 = 3(5 · 5) = 3 · 25 = 75 b.) Find –2x2 when x = –1. –2x2 = –2(–1)2 = –2(–1)(–1) = –2(1) = –2 6 The Product Rule Product Rule (applies to common bases only) am · an = am+n Example Simplify each of the following expressions. 32 · 34 = 32+4 = 36 = 3 · 3 · 3 · 3 · 3 · 3= 729 x4 · x5 = x4+5 = x9 z3 · z2 · z5= z3+2+5 = z10 (3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6 7 The Power Rule Power Rule (am)n = amn Example Simplify each of the following expressions. (23)3 = 23·3 = 29 = 512 (x4)2 = x4·2 = x8 8 The Power of a Product Rule Power of a Product Rule (ab)n = an · bn Example Simplify (5x2y)3. (5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3 9 The Power of a Quotient Rule Power of a Quotient Rule () a b Example Simplify n an = n b () a b n an = n b (5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3 10 The Power of a Quotient Rule Power of a Quotient Rule () a b n an = n b Example Simplify the following expression. 4 ( ) ( ) 2 4  p  p  3  = 3 r 3 3r   2 4 ( p) = 3 (r ) 2 4 4 3 4 (Power of product rule) p8 = 12 81r (Power rule) 11 The Quotient Rule Quotient Rule (applies to common bases only) am m−n = a an a≠0 Example Simplify the following expression. 9a 4b 7 =  9  a 4  b 7  4 −1 7−2 3 5   3 ( )( ) = a b = 3 a b 2     2  3  a  b  3ab Group common bases together 12 Zero Exponent Zero exponent a0 = 1, a ≠ 0 Note: 00 is undefined. Example Simplify each of the following expressions. 50 = 1 (xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1 –x0 = –(x0) = – 1 13