1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 12 Exponents and Polynomials

2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 12.1 Exponents

3. Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = 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. Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate each expression. a. 3 4 = 3 3 3 3 = 81 b. (–5) 2 = (– 5)(–5) = 25 c. –6 2 = – (6)(6) = –36 d. (2 4) 3 = (2 4)(2 4)(2 4) = 8 8 8 = 512 e. 3 4 2 = 3 16 = 48 Example

5. Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate each expressions for the given value of x. 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

6. Martin-Gay, Developmental Mathematics, 2e 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 a. 3 2 · 3 4 = 3 6 b. x 4 · x 5 = x 4+5 c. z 3 · z 2 · z 5 = z 3+2+5 d. (3y 2 )(– 4y 4 ) = 3 · y 2 (– 4) · y 4 = 3(– 4)(y 2 · y 4 ) = – 12y 6 = 3 2+4 = x 9 = z 10 The Product Rule for Exponents Example:

7. Martin-Gay, Developmental Mathematics, 2e 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 = 9 12 3 5 ∙ 3 7 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = 3 12 12 factors of 3, not 9. Add exponents. Common base not kept. 5 factors of 3. 7 factors of 3.

8. Martin-Gay, Developmental Mathematics, 2e 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.

9. Martin-Gay, Developmental Mathematics, 2e 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 Example: a. (2 3 ) 3 = 2 9 b. (x 4 ) 2 = x 8 = 2 3·3 = x 4·2 The Power Rule

10. Martin-Gay, Developmental Mathematics, 2e 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 Power of a Product Rule Example: = 5 3 · (x 2 ) 3 · y 3 = 125x 6 y 3 (5x 2 y) 3

11. Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. If n is a positive integer and a and c are real numbers, then Power of a Quotient Rule Example:

12. Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Quotient Rule for Exponents Example: If m and n are positive integers and a is a real number, then Group common bases together.

13. Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. a 0 = 1, as long as a is not 0. Note: 0 0 is undefined. Example: a. 5 0 = 1 b. (xyz 3 ) 0 = x 0 · y 0 · (z 3 ) 0 = 1 · 1 · 1 = 1 c. –x 0 = –(x 0 ) = – 1 Zero Exponent