Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 3 Exponents and Polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3.1 Exponents

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate each of the following expressions = = 81 (–5) 2 = (– 5)(–5)= 25 –6 2 = – (6)(6) = –36 (2 4) 3 = (2 4)(2 4)(2 4)= 8 8 8= = 3 4 4= 48 Evaluating Exponential Expressions Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate each of the following expressions. Evaluating Exponential Expressions Example a.) Find 3x 2 when x = 5. b.) Find –2x 2 when x = –1. 3x 2 = 3(5) 2 = 3(5 · 5)= 3 · 25 –2x 2 = –2(–1) 2 = –2(–1)(–1)= –2(1) = 75 = –2

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If m and n are positive integers and a is a real number, then a m · a n = a m+n 3 2 · 3 4 = 3 6 x 4 · x 5 = x 4+5 z 3 · z 2 · z 5 = z (3y 2 )(– 4y 4 )= 3 · y 2 (– 4) · y 4 = 3(– 4)(y 2 · y 4 ) = – 12y 6 = = x 9 = z 10 The Product Rule For example,

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Don’t forget that In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifying the exponential expression. 3 5 ∙ 3 7 = ∙ 3 7 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = factors of 3, not 9. Add exponents. Common base not kept. 5 factors of 3. 7 factors of 3.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Helpful Hint Don’t forget that if no exponent is written, it is assumed to be 1.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If m and n are positive integers and a is a real number, then (a m ) n = a mn For example, (2 3 ) 3 = 2 9 (x4)2(x4)2 = x 8 = 2 3·3 = x 4·2 The Power Rule

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If n is a positive integer and a and b are real numbers, then (ab) n = a n · b n The Power of a Product Rule For example, = 5 3 · (x 2 ) 3 · y 3 = 125x 6 y 3 (5x 2 y) 3

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If n is a positive integer and a and c are real numbers, then The Power of a Quotient Rule For example,

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The Quotient Rule For example, If m and n are positive integers and a is a real number, then Group common bases together.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. a 0 = 1, a ≠ 0 Note: 0 0 is undefined. For example, 5050 = 1 (xyz 3 ) 0 = x 0 · y 0 · (z 3 ) 0 = 1 · 1 · 1 = 1 –x0–x0 = –(x 0 ) = – 1 Zero Exponent