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Chapter 10
Exponential
Functions
Copyright © 2015, 2008 Pearson Education, Inc.
Section 10.1, Slide 1
10.1 Integer Exponents
Copyright © 2015, 2008 Pearson Education, Inc.
Section 10.1, Slide 2
Negative-integer exponent
Definition
If n is a counting number and b ≠ 0, then
b-n
1
 n
b
In words, to find b-n, take its reciprocal and change
the sign of the exponent.
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Section 10.1, Slide 3
Example: Simplifying Expressions
Involving Exponents
Simplify.
1. 5 2
2. 9b 7
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3. 31  4 1
Section 10.1, Slide 4
Solution
1
1
1. 5  2 
5
25
2
1
9
2. 9b  9  7  7
b
b
7
1 1 4 3 7
3. 3  4     
3 4 12 12 12
1
1
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Section 10.1, Slide 5
Negative-Integer Exponent in a
Denominator
If n is a counting number and b ≠ 0, then
1
n

b
bn
In words, to find 1 n , take its reciprocal and change
b
the sign of the exponent.
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Section 10.1, Slide 6
Example: Simplifying Expressions
Involving Exponents
Simplify.
1.
1
2 3
2.
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5
b 3
Section 10.1, Slide 7
Solution
1
1. 3  23  8
2
5
1
3

5


5b
2. 3
b
b 3
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Section 10.1, Slide 8
Example: Simplifying Power
Expressions
2 4
4
a
b
Simplify
.
5
7c
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Section 10.1, Slide 9
Solution
4a 2b 4 4c5b 4

5
7c
7a 2
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Section 10.1, Slide 10
Properties of Integer Exponents
If m and n are integers, b ≠ 0, and c ≠ 0, then
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Section 10.1, Slide 11
Simplifying Expressions Involving
Exponents
An expression involving exponents is simplified if
1. It includes no parentheses.
2. In any monomial, each variable or constant
appears as a base at most once.
3. Each numerical expression (such as 72) has been
calculated, and each numerical fraction has been
simplified.
4. Each exponent is positive.
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Section 10.1, Slide 12
Example: Simplifying Expressions
Involving Exponents
Simplify.
 3bc 
5 2
1.
 2b

2 2 3
c
4 7
 18b c 
2.  3 2 
 6b c 
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4
Section 10.1, Slide 13
Solution
1.
 3bc 
5 2
 2b

2 2 3
c
3 b c
2

2 b
3
2

5 2
 c 
2 3
2 3
9b 2c10
 6 6
8b c

9b
2( 6 ) 106
c
8
9b8c 4

8
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Section 10.1, Slide 14
Solution
4 7
4


18
b
c
4( 3) 7 2 4
2.  3 2    3b
c 
 6b c 
  3b c

b   c 
1 5 4
4
3
1 4
5 4
 34 b4c 20
4
b
 4 20
3c
b4

81c 20
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Section 10.1, Slide 15
Exponential function
Definition
An exponential function is a function whose
equation can be put into the form
f(x) = abx
where a ≠ 0, b > 0, and b ≠ 1. The constant b is
called the base.
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Section 10.1, Slide 16
Example: Evaluating Exponential
Functions
For f(x) = 3(2)x and g(x) = 5x, find the following.
1. f(3)
2. f(–4)
3. g(a + 3)
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4. g(2a)
Section 10.1, Slide 17
Solution
f(x) = 3(2)x and g(x) = 5x
3
f
(
3
)

3(2)
 3  8  24
1.
3 3
2. f (4)  3(2)  4 
2 16
4
3. g (a  3)  5a3  5a53  125(5)a
4. g (2a)  5   5
2a

2 a
 25
a
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Section 10.1, Slide 18
Exponential Functions
Warning
It is a common error to confuse exponential
functions such as E(x) = 2x with linear functions
such as L(x) = 2x.
For the exponential function, the variable x is the
exponent.
For the linear function, the variable x is a base.
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Section 10.1, Slide 19
Scientific Notation
Definition
A number is written in scientific notation if it has
the form N 10k , where k is an integer and the
absolute value of N is between 1 and 10 or is equal
to 1.
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Section 10.1, Slide 20
Example: Writing Numbers in Standard
Decimal Notation
Simplify.
1. 7  103
3
7

10
2.
4
9.48

10
3.
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Section 10.1, Slide 21
Solution
1. We simplify 7 103  7.0 103 by multiplying
7.0 by 10 three times and hence moving the
decimal point three places to the right:
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Section 10.1, Slide 22
Solution
2. Since
1
7 1
7
7 10  7  3   3  3
10
1 10 10
3
we see that we can simplify 7.0 103 by
dividing 7.0 by 10 three times and hence
moving the decimal point three places to
the left:
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Section 10.1, Slide 23
Solution
3. We divide 9.48 by 10 four times and hence
move the decimal point of 9.48 four places
to the left:
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Section 10.1, Slide 24
Converting from Scientific Notation to
Standard Decimal Notation
To write the scientific notation N  10 in standard
decimal notation, we move the decimal point on the
number N as follows:
• If k is positive, we multiply N by 10 k times; hence,
we move the decimal point k places to the right.
• If k is negative, we divide N by 10 k times; hence,
we move the decimal point k places to the left.
k
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Section 10.1, Slide 25
Example: Writing Numbers in
Scientific Notation
Write the number in scientific notation.
1. 845,000,000
2. 0.0000382
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Section 10.1, Slide 26
Solution
1. In scientific notation, we would have 8.45 10 ,
but what is k? If we move the decimal point of 8.45
eight places to the right, the result is 845,000,000.
So, k = 8 and the scientific notation is 8.45  108.
k
2. In scientific notation, we would have 3.82 10k ,
but what is k? If we move the decimal point of 3.82
five places to the left, the result is 0.0000382. So,
k = –5 and the scientific notation is 3.82 105.
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Section 10.1, Slide 27
Converting from Standard Decimal
Notation to Scientific Notation
To write a number in scientific notation, count the
number k of places that the decimal point needs to be
moved so that the absolute value of the new number
N is between 1 and 10 or is equal to 1:
• If the decimal point is moved to the left, then the
scientific notation is written as N 10k .
• If the decimal point is moved to the right, then the
k
scientific notation is written as N  10 .
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Section 10.1, Slide 28