Chapter Chapter 5 5 Section Section 2 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Chapter Chapter 55Section Section 22
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use 0 as an exponent.Use negative numbers as exponents.Use the quotient rule for exponents.Use combinations of rules.
Integer Exponents, and Quotient Rule
11
44
33
22
5.25.25.25.2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Integer Exponents and the Quotient RuleIn all earlier work, exponents were positive integers. Now, to
develop a meaning for exponents that are not positive integers, consider the following list.
Slide 5.2 - 3
4
3
2
2 16
2 8
2 4
Each time the exponent is reduced by 1, the value is divided
by 2 (the bases). Using this pattern, the list can be continued to smaller and smaller integers.
12 2 02 1 1 12
2 2 1
24
From the preceding list, it appears that we should define 20 as 1 and negative exponents as reciprocals.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Slide 5.2 - 4
Use 0 as an exponent.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
so that the product rule is satisfied. Check that the power rules are also valid for a 0 exponent. Thus we define a 0 exponent as follows.
Use 0 as an exponent.
The definitions of 0 and negative exponents must satisfy the rule for exponents from Section 5.1. For example if 60 = 1, then
and
Slide 5.2 - 5
0 2 2 26 6 1 6 6
For any nonzero real number a, a0 = 1.
Example: 170 = 1
0 2 0 2 26 6 6 6
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EXAMPLE 1 Using Zero Exponents
07
Solution:
1
01 7
Slide 5.2 - 6
Evaluate.
1
1
07
07
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Objective 22
Use negative numbers as exponents.
Slide 5.2 - 7
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use negative numbers as exponents.
Since and , we can deduce that 2−n should equal . Is the product rule valid in such a case? For example, if we
multiply
Slide 5.2 - 8
2 12
4 3 1
28
1
2n
2 2 2 2 06 6 6 6
The expression 6−2 behaves as if it were the reciprocal of 62: Their product is 1. The reciprocal of 62 is also , leading us to define 6−2 as . This is a particular case of the definition of negative exponents.
2
1
62
1
6
For any nonzero real number a and any integer n,
Example:
1.n
na
a
1.n
na
a
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EXAMPLE 2 Using Negative Exponents
24
Solution:
2
1
4
34
Slide 5.2 - 9
Simplify.
5 2
5
1
2 2
1
5
3
1
m
5 2
10 10
7
10
2
2
5
3
31
4
23
5
1 12 5
3 0m m
1
16
64
25
9
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use negative numbers as exponents.Consider the following:
Slide 5.2 - 10
For any nonzero numbers a and b and any integers m and n,
and
Therefore,
3 4 43
4 3 4 3 3
4
12 1 1 1 3 32
13 2 3 2 1 23
.
3 4
4 33.
2 3
2
m n
n m
a b
b a
-m ma b
=b a
Example: and5 4
4 5
3 2
2 3
3 3
4 5
5 4
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EXAMPLE 3
Solution:
Changing from Negative to Positive Exponents
3
2
3
5
Slide 5.2 - 11
Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.
27
25
2
5
4m
h k
3
3
3
2
2y
x
We cannot use this rule to change negative exponents to positive exponents if the exponents occur in a sum or difference of terms. For example,
would be written with positive exponents as .
2
3
5
3
5
2
4h
m k
32
32
x
y
9
6
8y
x
2 1
3
5 3
7 2
2
3
1 15 3
17
2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Use the quotient rule for exponents.
Slide 5.2 - 12
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We know that
Use the quotient rule for exponents.
Notice that the difference between the exponents, 5 − 3 = 2, this is the exponent in the quotient. This example suggests the quotient rule for exponents.
Slide 5.2 - 13
52
3
6 6 6 6 6 66
6 6 6 6.
For any nonzero real number a and any integer m and n,
(Keep the same base; subtract the exponents.)
Example:
.m
m nn
aa
a
88 4 4
4
55 5
5
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6
12
x
x
EXAMPLE 4
Solution:
Using the Quotient Rule
7
5
4
4
Slide 5.2 - 14
Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers.
5
7
4
4
7 54
4 9 3
5 10 2
8
8
m n
m n
6 ( 12)x 6x24
2
1
45 74 24
1
16
1 1 58
1
m n
4 5 9 10 3 28 m n
16
5
1
8mn
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.2 - 15
The product, quotient, and power rules are the same for positive and negative exponents.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 44
Use combinations of rules.
Slide 5.2 - 16
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5 Using Combinations of Rules
Solution:
Slide 5.2 - 17
Simplify. Assume that all variables represent nonzero real numbers.
225
6
y
24
3
3
3
224 4x x 29 2
3 4
3
3
x y
x y
8
3
3
3 8 33 53
2 2 24 4x x 1 2 2 24 x 3 44 x
464x
2 4
2
5
6
y
2
2 4
6
5 y 4
36
25y
9 3 4 2
4
3 x y
x y
6
3
3
y 3
729
y
243
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