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Chapter 4
Exponential
Functions
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 1
4.1 Properties of
Exponents
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 2
Exponent
Definition
For any counting number n,
b  b  b  b  ...b
n
n factors of b
We refer to bn as the power, the nth power of b, or
b raised to the nth power. We call b the base and n
the exponent.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 3
Properties of Exponents
If m and n are counting numbers, then
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 4
Example: Meaning of Exponential
Properties
1. Show that b2b3 = b5.
2. Show that bmbn = bm + n, where m and n are
counting numbers
n
n
b b

3. Show that    n , where n is a counting
c c
number and c ≠ 0.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 5
Solution
1. By writing b2b3 without exponents, we see
b2b3  (bb)(bbb)  bbbbb  b5
We can verify that this result is correct for
various constant bases by examining graphing
calculator tables for both y = x2x3 and y = x5.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 6
Solution
2. Write bmbn without exponents:
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Section 4.1, Slide 7
Solution
n
b

3. Write   , where c ≠ 0, without exponents:
c
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 8
Simplifying Expressions Involving
Exponents
An expression involving exponents is simplified if
1. It includes no parentheses.
2. Each variable or constant appears as a base as few
times as possible. For example, we write
x2x4 = x6.
3. Each numerical expression (such as 72) has been
calculated and each numerical fraction has been
simplified.
4. Each exponent is positive.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 9
Example: Simplifying Expressions
Involving Exponents
Simplify.
1. (2b2c3)5
7 6
3b c
3.
12b 2c5
2. (3b3c4)(2b6c2)
 24b c 
4. 
2 5 3
16
b
cd 

7 8
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
4
Section 4.1, Slide 10
Solution
1.
(2b c )  2 (b ) (c )
10 15
 32b c
2.
(3b3c 4 )(2b6c 2 )  (3  2)(b3b6 )(c 4c 2 )
2 3 5
5
2 5
3 5
 6b c
9 6
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 11
Solution
7 6
7 2 65
3
b
c
b
c
3.

2 5
12b c
4
4




24
b
c
3
b
c
4. 

2 5 3
3 
 16b c d   2d 
b5c

4
7 8
5 3
4
3b c 


 2d 
5 3 4
3 4
34 (b5 ) 4 (c3 ) 4

24 ( d 3 ) 4
81b 20c12

12
16d
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 12
Simplifying Expressions Involving
Exponents
Warning
The expressions 3b2 and (3b)2 are not equivalent
expressions:
3b2  3b  b
2
 3b    3b  3b   9b  b
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 13
Zero Exponent
Definition
For b ≠ 0,
b0 = 1
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Section 4.1, Slide 14
Negative integer exponent
Definition
If b ≠ 0 and n is a counting number, then
b-n
1
 n
b
In words, to find b-n, take its reciprocal and switch
the sign of the exponent.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 15
Negative Exponent in a Denominator
If b ≠ 0 and n is a counting number, then
1
n

b
bn
In words, to find 1 n , take its reciprocal and switch
b
the sign of the exponent.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 16
Example: Simplifying Expressions
Involving Exponents
Simplify.
1.
9b-7
5
2. 3
b
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3. 3-1 + 4-1
Section 4.1, Slide 17
Solution
1 9
1. 9b  9  7  7
b b
7
5
1
2. 3  5  3  5b3
b
b
1 1 4 3 7
3. 3  4     
3 4 12 12 12
1
1
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Section 4.1, Slide 18
Properties of Integer Exponents
If m and n are integers, b ≠ 0, and c ≠ 0, then
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Section 4.1, Slide 19
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 4.1, Slide 20
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 21
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 22
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 23
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 4.1, Slide 24
Solution
f(x) = 3(2)x and g(x) = 5x
1. f (3)  3(2)  3  8  24
3
3 3
2. f (4)  3(2)  4 
2 16
4
a 3
3. g (a  3)  5
 5 5  125(5)
4. g (2a)  5   5
2a
a 3

2 a
a
 25a
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 25
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 26
Scientific notation
Definition
A number is written in scientific notation if it
has the form N  10k , where k is an integers and
either –10 < N ≤ –1 or 1 ≤ N < 10.
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Section 4.1, Slide 27
Converting from Scientific Notation to
Standard Decimal Notation
To write the scientific notation N 10 in standard
decimal notation, we move the decimal point of the
number N as follows:
k
• 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 k places to the left.
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 28
Example: Converting to Standard
Decimal Notation
Write the number in standard decimal notation.
1. 3.462 10
5
2. 7.38 10
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4
Section 4.1, Slide 29
Solution
1. We multiply 3.462 by 10 five times; hence, we
move the decimal point of 3.462 five places to
the right:
2. We divide 7.38 by 10 four times; hence, we
move the decimal point of 7.38 four places to
the left:
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 30
Converting from Standard Decimal
Notation to Scientific Notation
To write a number in scientific notation, count the
number of places k that the decimal point must be
moved so the new number N meets the condition
–10 < N ≤ –1 or 1 ≤ N < 10:
• If the decimal point is moved to the left, then the
k
scientific notation is written as N 10 .
• If the decimal point is moved to the right, then the
k
scientific notation is written as N 10 .
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 31
Example: Converting to Scientific
Notation
Write the number in scientific notation.
1. 6,257,000,000
2. 0.00000721
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Section 4.1, Slide 32
Solution
1. In scientific notation, we would have
6.257 10k
We must move the decimal point of 6.257 nine
places to the right to get 6,257,000,000. So, k = 9
and the scientific notation is
6.257 109
Copyright © 2015, 2008, 2011 Pearson Education, Inc.
Section 4.1, Slide 33
Solution
2. In scientific notation, we would have
7.2110k
We must move the decimal point of 7.21 six
places to the left to get 0.00000721. So, k = –6
and the scientific notation is
7.21106
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Section 4.1, Slide 34