Download Define and use expressions of the form a 1/n

Chapter 8
Section 7
8.7 Using Rational Numbers as Exponents
Objectives
1
Define and use expressions of the form a1/n.
2
Define and use expressions of the form am/n.
3
Apply the rules for exponents using rational exponents.
4
Use rational exponents to simplify radicals.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Define and use expressions of the
form a1/n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-3
Define and use expressions of the form a1/n.
Now consider how an expression such as 51/2 should be defined, so
that all the rules for exponents developed earlier still apply. We define
51/2 so that
51/2 · 51/2 = 51/2 + 1/2 = 51 = 5.
This agrees with the product rule for exponents from Section 5.1. By
definition,
 5  5   5.
Since both 51/2 · 51/2 and 5  5 equal 5,
this would seem to suggest that 51/2 should equal 5.
Similarly, then 51/3 should equal 3 5.
Review the basic rules for exponents:
a m a n  a mn
a 
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
m n
 a mn
am
mn

a
an
Slide 8.7-4
Define and use expressions of the form a1/n.
a1/n
If a is a nonnegative number and n is a positive integer, then
a1/ n  n a .
Notice that the denominator of the rational exponent is the index
of the radical.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-5
EXAMPLE 1 Using the Definition of a1/n
Simplify.
Solution:
491/2
1000
811/4
1/3
 49
7
 3 1000
 10
 4 81
3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-6
Objective 2
Define and use expressions
of the form am/n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-7
Define and use expressions of the form am/n.
Now we can define a more general exponential expression, such as
163/4. By the power rule, (am)n = amn, so
 
 16
1/ 4 3
3/ 4
16
4

3
16
 23  8.
However, 163/4 can also be written as
3/ 4
16
 16

3 1/ 4
  4096 
1/ 4
 4 4096  8.
Either way, the answer is the same. Taking the root first involves
smaller numbers and is often easier. This example suggests the
following definition for a m/n.
am/n
If a is a nonnegative number and m and n are integers with n > 0, then
a
m/ n
 a
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
  a
1/ n m
n
m
.
Slide 8.7-8
EXAMPLE 2 Using the Definition of am/n
Evaluate.
Solution:
9
5/2
 9
8
5/3
 8
–27

 35
 243

 25
 32
1/ 2 5
1/ 3 5
2/3
   27

1/ 3 2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
   3  9
2
Slide 8.7-9
Using the definition of am/n.
Earlier, a–n was defined as
an 
1
an
for nonzero numbers a and integers n. This same result applies to
negative rational exponents.
a−m/n
If a is a positive number and m and n are integers, with n > 0, then
a
m / n

1
a
m/n
.
A common mistake is to write 27–4/3 as –273/4. This is incorrect. The
negative exponent does not indicate a negative number. Also, the negative
exponent indicates to use the reciprocal of the base, not the reciprocal of
the exponent.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-10
EXAMPLE 3 Using the Definition of a−m/n
Evaluate.
Solution:
36–3/2
1
 3/ 2
36
81–3/4
1
 3/ 4
81


Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1
 36 
1/ 2 3
1
81 
1/ 4 3
1
 3
6
1

216
1
 3
3
1

27
Slide 8.7-11
Objective 3
Apply the rules for exponents
using rational exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-12
Apply the rules for exponents using rational exponents.
All the rules for exponents given earlier still hold when the exponents
are fractions.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-13
EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents
Simplify. Write each answer in exponential form with only positive
exponents.
Solution:
71/ 3  7 2 / 3
 71/ 3 2 / 3
7
92 / 3
9 1/ 3
 92 / 31/ 3
9
 27 
 
 8 
5/ 3
31/ 2  32
35/ 2
5/ 3
27
 5/ 3
8
1/ 2 4/ 25/ 2
3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
27 


8 
35
 5
2
3
3
1/ 3 5
1/ 3 5
2/ 2
Slide 8.7-14
EXAMPLE 5 Using Fractional Exponents with Variables
Simplify. Write each answer in exponential form with only positive
exponents. Assume that all variables represent positive numbers.
Solution:
a

2 / 3 1/ 3 2 6
b c
r 2 / 3  r1/ 3
r 1
a 
 1/ 4 
b 
2/3
3
 a
 b   c 
2/ 3 6
1/ 3 6
2 6
 a12 / 3b 6 / 3c12
 a 4b 2 c12
 r 2/ 31/ 33/ 3
 r6/3
 r2
a 


b 
a6 / 3
 3/ 4
b
a2
 3/ 4
b
2/3 3
1/ 4 3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.7-15
Objective 4
Use rational exponents to simplify
radicals.
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Slide 8.7-16
Use rational exponents to simplify radicals.
Sometimes it is easier to simplify a radical by first writing it in
exponential form.
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Slide 8.7-17
EXAMPLE 6 Simplifying Radicals by Using Rational Exponents
Simplify each radical by first writing it in exponential form.
Solution:
4
12
2
 
6
x
3
 12
2 1/ 4

 x

3 1/ 6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
 121/ 2
 12
 x1/ 2
 x  x  0
2 3
Slide 8.7-18