chapter 8
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Chapter 8. Section 5. More Simplifying and Operations with Radicals. Simplify products of radical expressions. Use conjugates to rationalize denominators of radical expressions. Write radical expressions with quotients in lowest terms. 8.5. 2. 3. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 8 Section 5
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
More Simplifying and Operations with Radicals
Simplify products of radical expressions.
Use conjugates to rationalize denominators of radical expressions.
Write radical expressions with quotients in lowest terms.
8.5
2
3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
More Simplifying and Operations with Radicals
The conditions for which a radical is in simplest form were listed in the previous section. A set of guidelines to use when you are simplifying radical expressions follows:
Slide 8.5-3
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More Simplifying and Operations with Radicals (cont’d)
Slide 8.5-4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Simplify products of radical expressions.
Slide 8.5-5
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each product and simplify.
Solution:
2 8 20 2 5 3 3 2 2
2 2 2 4 5
2 2 2 4 5
2 2 2 2 5
4 2 5 2
4 2 10
2 3 2 2 2 5 3 3 5 3 2 2
6 11 10 6
11 9 6
Slide 8.5-6
EXAMPLE 1 Multiplying Radical Expressions
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each product and simplify.
Solution:
2 5 10 2
2 10 2 2 5 10 5 2
20 2 50 10
2 5 2 5 2 10
Slide 8.5-7
EXAMPLE 1 Multiplying Radical Expressions (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each product. Assume that x ≥ 0.
Solution:
2
5 3 2
4 2 5 2
2 x
2
25 2 5 3 3 2
24 2 2 4 2 5 5 222 2 2 x x
5 6 5 9
14 6 5
32 40 2 25
57 40 2
4 4 x x
Remember only like radicals can be combined!
Slide 8.5-8
EXAMPLE 2 Using Special Products with Radicals
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Using a Special Product with Radicals.
Example 3 uses the rule for the product of the sum and difference of two terms,
2 2.x y x y x y
Slide 8.5-9
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each product. Assume that 0.y
Solution:
3 2 3 2 4 4y y
2 2
3 2
3 4
1
2 2
4y
16y
Slide 8.5-10
EXAMPLE 3 Using a Special Product with Radicals
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Objective 2
Use conjugates to rationalize denominators of radical expressions.
Slide 8.5-11
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
The results in the previous example do not contain radicals. The pairs being multiplied are called conjugates of each other. Conjugates can be used to rationalize the denominators in more complicated quotients, such as
Use conjugates to rationalize denominators of radical expressions.
2.
4 3
Using Conjugates to Rationalize a Binomial Denominator
To rationalize a binomial denominator, where at least one of those terms is a square root radical, multiply numerator and denominator by the conjugate of the denominator.
Slide 8.5-12
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify by rationalizing each denominator. Assume that 0.t 3
2 55+3
2 5
2 5
2 55 2
3
2
2
3 2 5
2 5
3 2 5
4 5
3 2 5
1
3 2 5
2 5
2 5
5 3
2 5
2 2
2 5 5 6 3 5
2 5
5 5 11
4 5
5 5 11
1
5 5 11
11 5 5
Solution:
Slide 8.5-13
EXAMPLE 4 Using Conjugates to Rationalize Denominators
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify by rationalizing each denominator. Assume that 0.t
3
2 t
23
2 2
t
tt
2
2
3 2
2
t
t
3 2
4
t
t
Solution:
Slide 8.5-14
EXAMPLE 4 Using Conjugates to Rationalize Denominators (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Write radical expressions with quotients in lowest terms.
Slide 8.5-15
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write in lowest terms.
Solution:
5 3 15
10
5 3 3
10
3 3
2
Slide 8.5-16
EXAMPLE 5 Writing a Radical Quotient in Lowest Terms