Chapter 8 Section 7. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Using...
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Transcript of Chapter 8 Section 7. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Using...
Chapter 8 Section 7
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Using Rational Numbers as Exponents
Define and use expressions of the form a1/n.
Define and use expressions of the form am/n.
Apply the rules for exponents using rational exponents.
Use rational exponents to simplify radicals.
8.7
2
3
4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Define and use expressions of the form a1/n.
Slide 8.7-3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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,
Since both 51/2 · 51/2 and equal 5,
this would seem to suggest that 51/2 should equal
Similarly, then 51/3 should equal
5 5 5.
5 5
3 5.5.
Review the basic rules for exponents:
m n m na a a m
m nn
aa
a nm mna a
Slide 8.7-4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
a1/nIf a is a nonnegative number and n is a positive integer, then
1/ .n na a
Slide 8.7-5
Define and use expressions of the form a1/n.
Notice that the denominator of the rational exponent is the index of the radical.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify.
491/2
10001/3
811/4
Solution:
49 7
3 1000 10
4 81 3
Slide 8.7-6
EXAMPLE 1 Using the Definition of a1/n
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Define and use expressions of the form am/n.
Slide 8.7-7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
333/ 4 1/ 4 3416 16 16 2 8.
However, 163/4 can also be written as
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.
1/ 4 1/ 43/ 4 3 416 16 4096 4096 8.
am/nIf a is a nonnegative number and m and n are integers with n > 0, then
/ 1/ .mmm n n na a a
Slide 8.7-8
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate.
95/2
85/3
–272/3
Solution:
51/ 29 53
51/38 52
21/327 9
243
32
23
Slide 8.7-9
EXAMPLE 2 Using the Definition of am/n
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Earlier, a–n was defined as
for nonzero numbers a and integers n. This same result applies to negative rational exponents.
Using the definition of am/n.
1nn
aa
a−m/nIf a is a positive number and m and n are integers, with n > 0, then
//
1.m n
m na
a
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.
Slide 8.7-10
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
Evaluate.
36–3/2
81–3/4
3/ 2
1
36
3/ 4
1
81
31/ 2
1
36
3
1
6
1
216
31/ 4
1
81
3
1
3
1
27
Slide 8.7-11
EXAMPLE 3 Using the Definition of a−m/n
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Apply the rules for exponents using rational exponents.
Slide 8.7-12
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Apply the rules for exponents using rational exponents.
All the rules for exponents given earlier still hold when the exponents are fractions.
Slide 8.7-13
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
Simplify. Write each answer in exponential form with only positive exponents.
1/3 2/37 72/3
1/3
9
9
5/327
8
1/ 2 2
5/ 2
3 3
3
1/3 2 /37 7
2/3 1/39 9
5/3
5/3
27
8
51/3
51/3
27
8
5
5
3
2
1/ 2 4/ 2 5/ 23 2/ 23 3
Slide 8.7-14
EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.
Solution:
62/3 1/3 2a b c
2/3 1/3
1
r r
r
32/3
1/ 4
a
b
6 6 62/3 1/3 2a b c 12/3 6/3 12a b c 4 2 12a b c
2/3 1/3 3/3r 6/3r 2r
32/3
31/ 4
a
b
6/3
3/ 4
a
b
2
3/ 4
a
b
Slide 8.7-15
EXAMPLE 5 Using Fractional Exponents with Variables
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 4
Use rational exponents to simplify radicals.
Slide 8.7-16
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use rational exponents to simplify radicals.
Sometimes it is easier to simplify a radical by first writing it in exponential form.
Slide 8.7-17
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify each radical by first writing it in exponential form.
4 212
36 x
1/ 4212 1/ 212 12
1/ 63x 1/ 2x 0x x
Solution:
2 3
Slide 8.7-18
EXAMPLE 6 Simplifying Radicals by Using Rational Exponents