chapter 8
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Chapter 8 Section 4
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Rationalizing the Denominator
Rationalize denominators with square roots.
Write radicals in simplified form.
Rationalize denominators with cube roots.
8.4
2
3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Rationalize denominators with square roots.
Slide 8.4-3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Rationalize denominators with square roots.
It is easier to work with a radical expression if the denominators do not contain any radicals.
1 1 222 2
22
2 .2
This process of changing the denominator from a radical, or irrational number, to a rational number is called rationalizing the denominator.The value of the radical expression is not changed; only the form is changed, because the expression has been multiplied by 1 in the form of
Slide 8.4-4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Rationalize each denominator.
Solution:
1824
6182 6 6
18 62 6
18 612
168
2162 2 2
16 22 2
16 24
4 2
3 62
Slide 8.4-5
EXAMPLE 1 Rationalizing Denominators
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Write radicals in simplified form.
Slide 8.4-6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write radicals in simplified form.
Conditions for Simplified Form of a Radical
1. The radicand contains no factor (except 1) that is a perfect square (in dealing with square roots), a perfect cube (in dealing with cube roots), and so on.
2. The radicand has no fractions.
3. No denominator contains a radical.
Slide 8.4-7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
5 .18
518
8
518
181
5 1818
5 9 2
18
5 9 2
18
3 5 218
3 1018
106
Slide 8.4-8
EXAMPLE 2 Simplifying a Radical
Simplify
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify
Solution:
1 5.2 6
1 52 6
5
12
512
35
2 3 3
5 36
156
Slide 8.4-9
EXAMPLE 3 Simplifying a Product of Radicals
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify. Assume that p and q are positive numbers.
Solution:
5pq
5 qpq q
5pqq
Slide 8.4-10
EXAMPLE 4 Simplifying Quotients Involving Radicals
357
pq
2 2357p q
2 257
77
p q
2 2 357
p q
2 257p q
2 257p q
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Rationalize denominators with cube roots.
Slide 8.4-11
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Rationalize each denominator.
Solution:
35 6
3
3
23
3
3
3 , 04x
x
2
3
3
23
3 656 6
3 2
3 3
5 6
6
3 1806
2
3
3
23
3 323 3
3 2
3 3
2 3
3
3 183
3 2 2
3 23 2
3 4
4
34
xx x
3 2
3 3 3
3 16
4
x
x
23 3 2 84x
x
3 23 8 6
4x
x
3 262xx
Slide 8.4-12
EXAMPLE 5 Rationalizing Denominators with Cube Roots
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