10.5 Multiplying and Dividing Radical Expressions
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Transcript of 10.5 Multiplying and Dividing Radical Expressions
10.5 Multiplying and Dividing Radical Expressions
Multiply radical expressions.
Objective 1
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We multiply binomial expressions involving radicals by using the FOIL method from Section 4.5. Recall that this method refers to multiplying the First terms, Outer terms, Inner terms, and Last terms of the binomials.
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Multiply radical expressions.
Multiply, using the FOIL method.
2 3 1 5 2 2 5 3 15
F O I L
4 5 4 5 16 4 5 4 5 5 11
This is a difference of squares.
213 2
13 2 13 2 13 4
17 4 13
13 2 13 2
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CLASSROOM EXAMPLE 1 Multiplying Binomials Involving Radical Expressions
Solution:
3 34 7 4 7 16 3 4 7 3 4 7 3 2 7
0 and 0
r s r s
r s
2 2r s
r s
316 49
Difference of squares
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CLASSROOM EXAMPLE 1 Multiplying Binomials Involving Radical Expressions (cont’d)
Multiply, using the FOIL method.
Solution:
Rationalize denominators with one radical term.
Objective 2
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Rationalizing the Denominator
A common way of “standardizing” the form of a radical expression is to have the denominator contain no radicals. The process of removing radicals from a denominator so that the denominator contains only rational numbers is called rationalizing the denominator. This is done by multiplying y a form of 1.
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Rationalize denominators with one radical.
Rationalize each denominator.
511
Multiply the numerator and denominator by the denominator. This is in effect multiplying by 1.
51
11111
5 11
11
5 65 5
5 65
5
5 30
5 30
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CLASSROOM EXAMPLE 2 Rationalizing Denominators with Square Roots
Solution:
818 188
18 18
8 1818
8 9 218
24 218
4 23
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CLASSROOM EXAMPLE 2 Rationalizing Denominators with Square Roots (cont’d)
Rationalize the denominator.
Solution:
Simplify the radical.
845
845
4 29 5
2 23 5
52 23 5 5
2 103 5
2 1015
Factor.
Product Rule
Multiply by radical in denominator.
Product Rule
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CLASSROOM EXAMPLE 3 Rationalizing Denominators in Roots of Fractions
Solution:
6
7
200 , 0k yy
6
7
200k
y
3 2
6
100 2 ( )k
y y
3
3
10 2ky y
3
3
10 2y
yy
ky
3
4
10 2k yy
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CLASSROOM EXAMPLE 3 Rationalizing Denominators in Roots of Fractions (cont’d)
Simplify the radical.
Solution:
Simplify. 3
1532
3
3
1532
3
33
158 4
3
3
152 4
3
3
3
3
152 24
2
3
3
3302 8
304
42
6 , 0, 0y y ww
4
24
6y
w
244
2 244
6y
w
w
w
24 6yw
w
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CLASSROOM EXAMPLE 4 Rationalizing Denominators with Cube and Fourth Roots
Solution:
Rationalize denominators with binomials involving radicals.
Objective 3
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To rationalize a denominator that contains a binomial expression (one that contains exactly two terms) involving radicals, such as
we must use conjugates. The conjugate ofIn general, x + y and x − y are conjugates.
31 2
1 2 is 1 2.
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Rationalize denominators with binomials involving radicals.
Rationalizing a Binomial DenominatorWhenever a radical expression has a sum or difference with square root radicals in the denominator, rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
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Rationalize denominators with binomials involving radicals.
Rationalize the denominator.
42 3
4
2
3
2 33
2
4
4
2
3
3
34 2
1
4(2 3), or 8 4 3
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CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators
Solution:
Rationalize the denominator.
72 13
2 13
3
7
12 213
7 2 13
2 13
7 2 13
11
7 2 13
11
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CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators (cont’d)
Solution:
3 52 7
2 72 7
3 52 7
6 21 10 352 7
6 21 10 355
6 21 10 35
5
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CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators (cont’d)
Rationalize the denominator.
Solution:
2 ,
, 0, 0k zk z k z
2 k z
k z k z
2 k z
k z
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CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators (cont’d)
Rationalize the denominator.
Solution:
Write radical quotients in lowest terms.
Objective 4
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Write the quotient in lowest terms.
24 36 716 12 2 3 7
16
Factor the numerator and denominator.
4
4
3 2 3 7
4
3 2 3 7
4
Divide out common factors.
6 9 7or 4
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CLASSROOM EXAMPLE 6 Writing Radical Quotients in Lowest Terms
Solution:
Be careful to factor before writing a quotient in lowest terms.
22 32 , 06
x x xx
Factor the numerator.
Divide out common factors.
2 4 26
x xx
2 1 2 2
6
x
x
1 2 22
2 3
x
x
1 2 23
Product rule
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CLASSROOM EXAMPLE 6 Writing Radical Quotients in Lowest Terms (cont’d)
Write the quotient in lowest terms.
Solution: