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

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

Chapter 12

Exponents and Polynomials

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

12.1

Exponents

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.

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.

Martin-Gay, Developmental Mathematics, 2e 44

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

Evaluate each expression.

a. 34 = 3 • 3 • 3 • 3 = 81

b. (–5)2 = (– 5)(–5) = 25

c. –62 = – (6)(6) = –36

d. (2 • 4)3 = (2 • 4)(2 • 4)(2 • 4) = 8 • 8 • 8 = 512

e. 3 • 42 = 3 • 16 = 48

Example

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 3x2 when x = 5.

b. Find –2x2 when x = –1.

3x2 = 3(5)2 = 3(5 · 5) = 3 · 25

–2x2 = –2(–1)2 = –2(–1)(–1) = –2(1)

= 75

= –2

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

am · an = am+n

a. 32 · 34 = 36

b. x4 · x5 = x4+5

c. z3 · z2 · z5 = z3+2+5

d. (3y2)(– 4y4) = 3 · y2 (– 4) · y4 = 3(– 4)(y2 · y4) = – 12y6

= 32+4

= x9

= z10

The Product Rule for Exponents

Example:

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.

35 ∙ 37 = 912

35 ∙ 37 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3

= 312 12 factors of 3, not 9.

Add exponents.

Common base not kept.

5 factors of 3. 7 factors of 3.

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.

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

(am)n = amn

Example:

a. (23)3 = 29

b. (x4)2 = x8

= 23·3

= x4·2

The Power Rule

Martin-Gay, Developmental Mathematics, 2e 1010

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 = an · bn

Power of a Product Rule

Example:

= 53 · (x2)3 · y3 = 125x6 y3(5x2y)3

Martin-Gay, Developmental Mathematics, 2e 1111

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

ac

n

an

cn,c 0

Example:

p4

4

p4

44

Martin-Gay, Developmental Mathematics, 2e 1212

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

am

anam n,a 0

2

74

3

9

ab

ba 533 ba))((3 2714 ba

2

74

3

9

b

b

a

a

Group common bases together.

Martin-Gay, Developmental Mathematics, 2e 1313

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

a0 = 1, as long as a is not 0.

Note: 00 is undefined.

Example:

a. 50 = 1

b. (xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1

c. –x0 = –(x0) = – 1

Zero Exponent

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