Section 1 Part 1 Chapter 5. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 6 5 3 4 Integer Exponents – Part 1 Use the product rule.

Section 1Part 1

Chapter 5

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Objectives

2

6

5

3

4

Integer Exponents – Part 1

Use the product rule for exponents.

Define 0 and negative exponents.

Use the quotient rule for exponents.

5.1


We use exponents to write products of repeated factors. For example,

25 is defined as 2 • 2 • 2 • 2 • 2 = 32.

The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 25 is called an exponential or a power. We read 25 as “2 to the fifth power” or “2 to the fifth.”

Integer Exponents

Slide 5.1- 3



Objective 1

Slide 5.1- 4


Product Rule for ExponentsIf m and n are natural numbers and a is any real number, then

am • an = am + n.

That is, when multiplying powers of like bases, keep the same base and add the exponents.

Slide 5.1- 5


Be careful not to multiply the bases. Keep the same base and add the exponents.


Apply the product rule, if possible, in each case.

a) m8 • m6

b) m5 • p4

c) (–5p4) (–9p5)

d) (–3x2y3) (7xy4)

= m8+6 = m14

Cannot be simplified further because the bases m and p are not the same. The product rule does not apply.

= 45p9 = (–5)(–9)(p4p5) = 45p4+5

= –21x3y7 = (–3)(7) x2xy3y4 = –21x2+1y3+4

Slide 5.1- 6

CLASSROOM EXAMPLE 1

Using the Product Rule for Exponents

Solution:



Objective 2

Slide 5.1- 7


Zero ExponentIf a is any nonzero real number, then

a0 = 1.

Slide 5.1- 8


The expression 00 is undefined.


Evaluate.

290

(–29)0

–290

80 – 150

= 1

= 1

= – (290) = –1

= 1 – 1 = 0

Slide 5.1- 9

CLASSROOM EXAMPLE 2

Using 0 as an Exponent

Solution:


Negative Exponent

For any natural number n and any nonzero real number a,

1 .nn

aa

A negative exponent does not indicate a negative number; negative exponents lead to reciprocals.

22

1 13 Not negative

3 9

22

1 13 Negati

9v

3e

Slide 5.1- 10



Write with only positive exponents.

6-5

(2x)-4 , x ≠ 0

–7p-4, p ≠ 0

Evaluate 4-1 – 2-1.

5

1

6

4

1, 0

2x

x

4

17p

4

7, 0p

p

1 1

4 2

1 2

4 4 1

4

Slide 5.1- 11

CLASSROOM EXAMPLE 3

Using Negative Exponents

Solution:


Evaluate.

3

1

4

3

1319

3

114

3

11

4

341

1 34 64

3

1

3

9

3

1 1

3 9

3

1 9

3 1

1 9

27 1

9

27

1

3

Slide 5.1- 12

CLASSROOM EXAMPLE 4

Using Negative Exponents

Solution:


Special Rules for Negative Exponents

If a ≠ 0 and b ≠ 0 , then

and1 nna

a

.n m

m n

a b

b a

Slide 5.1- 13




Objective 3

Slide 5.1- 14


Quotient Rule for Exponents

If a is any nonzero real number and m and n are integers, then

That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator.

.m

m nn

aa

a

Slide 5.1- 15


Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses.


Apply the quotient rule, if possible, and write each result with only positive exponents.

8

13

m

m

6

8

5

5

8 13 55

1, 0m m m

m

6 ( 8) 6 8 25 5 5 , or 25

3

5, 0

xy

y Cannot be simplified because the bases x and y are

different. The quotient rule does not apply.

Slide 5.1- 16

CLASSROOM EXAMPLE 5

Using the Quotient Rule for Exponents

Solution:

Section 1 Part 1 Chapter 5. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 6 5 3 4 Integer Exponents – Part 1 Use the product rule.

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