Download Section 10.2

Section 10.2
Rational Exponents
Copyright © 2011 Pearson Education, Inc.
Definition of Rational Exponents
Definition of Rational Expressions
Introduction
• How should we define b , n is a counting number?
m n
• Exponent property b   b mn
• (–3)2 = 9 and 32 = 9 suggest (32) ½ = 9 ½ = 3
• The nonnegative number 3 is the principal second
root, or principle square root, of 9, written 9
1n
3
1
3


1
3
• If m = 3, n = 3:  ( 8)   8  81  8


1
3
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 2
Definition of Rational Exponents
Definition of Rational Expressions
Introduction
• 23 = 8
• Suggests that a good meaning of 81 3 is 2
• The number 2 is called the third root, or cube root,
3
of 8 written 8.
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 3
Definition of Rational Exponents
Definition of Rational Expressions
Definition
For the counting number n, where n ≠ 1,
• If n is odd, then b1 n is the number whose nth power
is b, and we call b1 n the nth root of b.
1n
• If n is even and b  0 , then b is the nonnegative
1n
number whose nth power is b, and we call b the
principle n root of b.
• If n is even and b < 0, then b1 n is not a real number
n
1n
b may be represented by b.
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 4
Simplifying Expressions Involving Rational Exponents
Definition of Rational Expressions
Example
Simplify.
12
2. 64
14
5.  16
1. 25
4. 16
3.  64 
13
13
6.  16 
14
14
Solution
1. 251 2  5, since 52  25
13
2. 64
 4, since 4  64
3
3.  64 
13
 4, since  4   64
3
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 5
Simplifying Expressions Involving Rational Exponents
Definition of Rational Expressions
Solution Continued
14
4. 16
 2, since 2  16
4
5.  161 4   161 4   2
6.  16  is not a real number, since the fourth
power of any real number is nonnegative.
• Graphing calculator checks
problems 1, 2 and 3
14
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 6
Definition: Rational Exponent
Definition of Rational Expressions
Definition
For the counting numbers m and n, where n ≠ 1 and b
1n
is any real number for which b is a real number,
b
b
m
 
 b
n
m
1
n

n
1
b
m
m
 b
m

1
n
, b0
n
A power of the form b m n or b  m n is said to have a
rational exponent.
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 7
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Example
Simplify.
1. 253 2
2.  27 
23
3. 32
2 5
4.  8 
5 3
Solution
32
1. 25
  25
2.  27 

12 3
23

 53  125

13 2
  27 
  3  9
2
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 8
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Solution Continued
3. 32
2 5
4.  8 
1
1
1 1
 25 
 2
2
15
32
32
  2 4
5 3
1


53
 8 
1
  8 
13 5
1
1


5
32
 2 
• Graphing calculator checks
problems 1, 2 and 3
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 9
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Example
For f  x   64 , g  x   3 16  , and h  x   5  9  ,
2
3
1





find the following:1. f   2. g   3. h   
 3
 4
 2
x
x
x
Solution
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 10
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Solution Continued
Properties
If m and n are real rational numbers and b and c are
any real number for which bm, bn and cn are real
numbers
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 11
Properties of Rational Expressions
Properties of Rational Expressions
Properties Continued
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 12
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Example
Simplify. Assume that b is positive.
27
3
2
b
6
1.  4b 
2. 3 7
b
Solution
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 13
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued
Example
Simplify. Assume that b is positive
2 25
 32b 
2 3 13
1. b b
2.  12 
 b 
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 14
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 15
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 16
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued
Example
Simplify. Assume that b and
c are constants.
81b c 
 27b c 
6 20 1 2
12 9 2 3
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 17
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 18
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued
Copyright © 2011 Pearson Education, Inc.
Section 10.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 19