Download 5.5 Integer Exponents and the Quotient Rule

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5.5 – Slide 1
Chapter 5
Exponents and Polynomials
Copyright © 2010 Pearson Education, Inc. All rights reserved.
5.5 – Slide 2
5.5
Integer Exponents and the
Quotient Rule
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5.5 – Slide 3
5.5 Integer Exponents and the Quotient Rule
Objectives
1.
2.
3.
4.
Use 0 as an exponent.
Use negative numbers as exponents.
Use the quotient rule for exponents.
Use combinations of rules.
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5.5 – Slide 4
5.5 Integer Exponents and the Quotient Rule
Using 0 as an Exponent
Zero Exponent
For any nonzero real number a,
a0 = 1.
Example: 170 = 1
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5.5 – Slide 5
5.5 Integer Exponents and the Quotient Rule
Using 0 as an Exponent
Example 1 Evaluate.
(a) 380 = 1
(b) (–9)0 = 1
(c) –90 = –1(9)0 = –1(1) = –1
(d) x0 = 1
(e) 5x0 = 5·1 = 5
(f) (5x)0 = 1
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5.5 – Slide 6
5.5 Integer Exponents and the Quotient Rule
Using Negative Numbers as Exponents
Negative Exponents
For any nonzero real number a and any integer n,
1
n
a  n.
a
1 1
Example: 32  2  .
3 9
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5.5 – Slide 7
5.5 Integer Exponents and the Quotient Rule
Using Negative Numbers as Exponents
Example 2
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
1
1
–3
(a) 9  3 
9
729
3
3
5
5
1 4
(b)       64
4 1
Notice that we can change the
base to its reciprocal if we also
change the sign of the exponent.
 2   3   243
(c)     
32
3 2
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5.5 – Slide 8
5.5 Integer Exponents and the Quotient Rule
Using Negative Numbers as Exponents
Example 2 (concluded)
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
1 1
(d) 6  3  
6 3
1 2
 
6 6
1

6
1
1
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3
3 x
(e) 4 
1 x 4
x
x4
 3x 4
4
=1
5.5 – Slide 9
5.5 Integer Exponents and the Quotient Rule
Using Negative Numbers as Exponents
CAUTION
A negative exponent does not indicate a negative
number. Negative exponents lead to reciprocals.
Expression
Example
a–n
1 1
2  3
2
8
–a–n
23  
3
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1
1


23
8
Not negative
Negative
5.5 – Slide 10
5.5 Integer Exponents and the Quotient Rule
Using Negative Numbers as Exponents
Changing from Negative to Positive Exponents
For any nonzero numbers a and b and any integers m and n,
m
n
a
b
 m
n
b
a
5
Examples:
4
3
2
 5
4
2
3
 a
and  
 b
3
m
m
 b
  .
 a
3
4
5
and      .
5
4
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5.5 – Slide 11
5.5 Integer Exponents and the Quotient Rule
Using Negative Numbers as Exponents
CAUTION
a m bn
Be careful. We cannot use the rule  n  m to
b
a
change negative exponents to positive exponents if the
exponents occur in a sum or difference of terms. For
example,
52  31
1 1
7  23

2
5 3.
would be written with positive exponents as
1
7 3
2
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5.5 – Slide 12
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
Quotient Rule for Exponents
For any nonzero number a and any integers m and n,
am
m n

a
.
n
a
(Keep the same base and subtract the exponents.)
58
Example: 6  586 =52 =25.
5
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5.5 – Slide 13
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
CAUTION
58
A common error is to write 6  186  12. This is
5
incorrect.
By the quotient rule, the quotient must have the same
base, 5, so
58
86
2

5
=5
.
6
5
We can confirm this by using the definition of
exponents to write out the factors:
58 55555555

.
6
555555
5
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5.5 – Slide 14
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
Example 3
Simplify. Assume that all variables represent nonzero real
numbers.
1
34
4 6
2

(a) 6  3  3
32
3
y 4
(b) 9  y 4( 9)  y 49 y 5
y
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5.5 – Slide 15
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
Example 3 (continued)
Simplify. Assume that all variables represent nonzero real
numbers.
24 ( z  a ) 7
(c) 5
2 ( z  a)6
24 ( z  a ) 7
 5 
2 ( z  a)6
 24( 5)  ( z  a)76
 29  ( z  a)
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5.5 – Slide 16
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
Example 3 (concluded)
Simplify. Assume that all variables represent nonzero real
numbers.
5x 3 y8
5 x 3 y 8
(d) 2 4 6  2  4  6
3 x y 3
x y
 5  32  x 34  y86
 5  9x 7 y 2
45y 2
 7
x
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5.5 – Slide 17
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
Definitions and Rules for Exponents
For any integers m and n:
Product rule
am · an = am+n
Zero exponent
a0 = 1 (a ≠ 0)
Negative exponent
Quotient rule
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a
n
1
 n
a
am
m n

a
an
(a  0)
5.5 – Slide 18
5.5 Integer Exponents and the Quotient Rule
Using the Quotient Rule for Exponents
Definitions and Rules for Exponents (concluded)
For any integers m and n:
Power rules
(a) (am)n = amn
(b) (ab)m = ambm
m
 a
am
(c)
 b   b m
Negative-to-Positive
Rules
a m bn
 m
n
b
a
 a
 b 
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m
(b  0)
(a, b  0)
 b
 
 a
m
5.5 – Slide 19
5.5 Integer Exponents and the Quotient Rule
Using Combinations of Rules
Example 4
Simplify each expression. Assume all variables represent
nonzero real numbers.
6
(23 ) 2
2
(a)
 6
26
2
 266
 20
1
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5.5 – Slide 20
5.5 Integer Exponents and the Quotient Rule
Using Combinations of Rules
Example 4 (continued)
Simplify each expression. Assume all variables represent
nonzero real numbers.
(3y ) 4 (3y ) 2 (3 y ) 42
(b)

1
(3y )
(3 y ) 1
 (3 y)6( 1)
 (3 y)7
 37 y 7
 2187 y 7
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5.5 – Slide 21
5.5 Integer Exponents and the Quotient Rule
Using Combinations of Rules
Example 4 (concluded)
Simplify each expression. Assume all variables represent
nonzero real numbers.
2
2
 5a   2 b 
(c)  1 4    3 
 2 b   5a  2
 a3  b4 
 1 
 2 5 
3
1
4
a 6 b8

(10) 2
a 6 b8

100
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5.5 – Slide 22
5.5 Integer Exponents and the Quotient Rule
Using Combinations of Rules
Note
Since the steps can be done in several different orders,
there are many equally correct ways to simplify
expressions like those in Example 4.
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5.5 – Slide 23