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Section 4.1
Properties of Exponents
Definition: Exponent
Definition of an Exponent
Definition
For any counting number n,
b  bbb
n
b
N factors of b
We refer to bn at the power; the nth power of b, or b
raised to the nth power.
We call b the base and n
the exponent.
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 2
Definition: Exponent
Definition of an Exponent
Definition
Two powers of b have specific names. We refer to b2
as the square of b or b squared. We refer to b3 as the
cube of b or b cubed.
Clarify
For –bn, we compute bn before finding the opposite.
For –24, the base is 2, not –2. If we want the base –2
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 3
Definition: Exponent
Definition of an Exponent
Calculator
• Use a graphing calculator to check both
computations
• To find –24, press (–) 2 ^ 3 ENTER
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 4
Properties of Exponents
Properties of Exponent
Properties
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 5
Properties of Exponents
Properties of Exponent
Example
Show that b5b3 = b5.
Solution
• Writing b5b3 without exponents, we see that
• Use calculator to verify by using various bases and
examining the table
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 6
Properties of Exponents
Properties of Exponent
Solution Continued
Example
Show that bmbn = bm+n, where m and n are counting
numbers.
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 7
Properties of Exponents
Properties of Exponent
Solution
• Write bmbn without exponents:
Example
n
n
b
b
Show that    n , n is a counting number and
c c
c ≠ 0.
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 8
Properties of Exponents
Properties of Exponent
Solution
n
b

• Write   without exponents:
c
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 9
Simplifying Expressions Involving Exponents
Simplifying Expressions Involving Exponents
Property
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.
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 10
Simplifying Expressions Involving Exponents
Simplifying Expressions Involving Exponents
Example
Simplify.
1.  2b c

 2b c 
2 3 5
2.  3b3c 4
2 2
3b 7 c 6
3.
2 5
12b c
 24b c 
4. 
2 5 3
 16b c d 
7 8
Section 4.1
4
Lehmann, Intermediate Algebra, 4ed
Slide 11
Simplifying Expressions Involving Exponents
Simplifying Expressions Involving Exponents
Solution
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 12
Simplifying Expressions Involving Exponents
Simplifying Expressions Involving Exponents
Solution Continued
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 13
Simplifying Expressions Involving Exponents
Simplifying Expressions Involving Exponents
Warning
• 3b2 and (3b)2 are not equivalent
• 3b2 base is b, and (3b)2 base is the 3b
• Typical error looks like
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 14
Simplifying Expressions Involving Exponents
Zero as an Exponent
Introduction
m
b
mn
0
What is the meaning of b ? The property n  b
b
is to be true for m = n, then
bn
nn
0
1 n  b  b , b  0
b
So, a reasonable definition of b0 is 1.
Definition
For b ≠ 0,
b0 = 1
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 15
Simplifying Expressions Involving Exponents
Zero as an Exponent
Illustration
• 70 = 1, (–3)0 = 1, and (ab)0 = 1, where ab ≠ 0
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 16
Negative Exponents
Negative Exponents
Introduction
If n is a negative integer, what is the meaning of bn?
What is them meaning of a negative exponent? If the
property b n  b mn is true for m = 0, then
b
0
1 b
0 n
n


b

b
, b0
n
n
b
b
1
–n
So, we would define b to be n .
b
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 17
Negative Integer Exponents
Negative Exponents
Definition
If b ≠ 0 and n is a counting number, then
1
b  n
b
In words: To find b–n, take its reciprocal and switch
the sign of the exponent.
n
Illustration
For example 32  12  1 and b 5  15
3 9
b
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 18
Negative Exponents
Negative Exponents
Introduction
1
We write  n in another form, where b ≠ 0 and n is a
b
counting number:
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 19
Negative Exponents
Negative Exponents
Definition
If b ≠ 0 and n is a counting number, then
1
n
b
n
b
In words: To find 1 n , take its reciprocal and switch t
b
he sign of the exponent.
Example
1
1
4
For example, 4  28  16 and 8  b8 .
2
b
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 20
Simplifying More Expressions Involving Exponents
Simplify More Expressions Involving Exponents
Example
Simplify.
1. 9b
7
5
2. 3
b
1
3. 3  4
1
Solution
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 21
Properties of Integer Exponents
Simplify More Expressions Involving Exponents
Properties
Section 4.1
Lehmann, Intermediate Algebra, 4ed
Slide 22