10.5 multiplying and dividing radical expressions
TRANSCRIPT
10.5 Multiplying and Dividing Radical Expressions
Multiply radical expressions.
Objective 1
Slide 10.5- 2
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.
Slide 10.5- 3
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.
2
13 2
13 2 13 2 13 4
17 4 13
13 2 13 2
Slide 10.5- 4
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 2
r s
r s
316 49
Difference of squares
Slide 10.5- 5
CLASSROOM EXAMPLE 1 Multiplying Binomials Involving Radical Expressions (cont’d)
Multiply, using the FOIL method.
Solution:
Rationalize denominators with one radical term.
Objective 2
Slide 10.5- 6
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.
Slide 10.5- 7
Rationalize denominators with one radical.
Rationalize each denominator.
5
11
Multiply the numerator and denominator by the denominator. This is in effect multiplying by 1.
5
1
11
111
5 11
11
5 6
5 5
5 6
5
5
5 30
5 30
Slide 10.5- 8
CLASSROOM EXAMPLE 2 Rationalizing Denominators with Square Roots
Solution:
8
18
188
18 18
8 18
18
8 9 2
18
24 2
18
4 2
3
Slide 10.5- 9
CLASSROOM EXAMPLE 2 Rationalizing Denominators with Square Roots (cont’d)
Rationalize the denominator.
Solution:
Simplify the radical.
8
45
8
45
4 2
9 5
2 2
3 5
52 2
3 5 5
2 10
3 5
2 10
15Factor.
Product Rule
Multiply by radical in denominator.
Product Rule
Slide 10.5- 10
CLASSROOM EXAMPLE 3 Rationalizing Denominators in Roots of Fractions
Solution:
6
7
200, 0
ky
y
6
7
200k
y
3 2
6
100 2 ( )k
y y
3
3
10 2k
y y
3
3
10 2
y
y
y
k
y
3
4
10 2k y
y
Slide 10.5- 11
CLASSROOM EXAMPLE 3 Rationalizing Denominators in Roots of Fractions (cont’d)
Simplify the radical.
Solution:
Simplify. 3
15
32
3
3
15
32
3
33
15
8 4
3
3
15
2 4
3
3
3
3
15
2 24
2
3
3
330
2 8
30
4
42
6, 0, 0yy w
w
4
24
6y
w
244
2 244
6y
w
w
w
24 6yw
w
Slide 10.5- 12
CLASSROOM EXAMPLE 4 Rationalizing Denominators with Cube and Fourth Roots
Solution:
Rationalize denominators with binomials involving radicals.
Objective 3
Slide 10.5- 13
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.
3
1 21 2 is 1 2.
Slide 10.5- 14
Rationalize denominators with binomials involving radicals.
Rationalizing a Binomial Denominator
Whenever 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.
Slide 10.5- 15
Rationalize denominators with binomials involving radicals.
Rationalize the denominator.
4
2 3
4
2
3
2 33
2
4
4
2
3
3
34 2
1
4(2 3), or 8 4 3
Slide 10.5- 16
CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators
Solution:
Rationalize the denominator.
7
2 13
2 13
3
7
12 213
7 2 13
2 13
7 2 13
11
7 2 13
11
Slide 10.5- 17
CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators (cont’d)
Solution:
3 5
2 7
2 7
2 7
3 5
2 7
6 21 10 35
2 7
6 21 10 35
5
6 21 10 35
5
Slide 10.5- 18
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
Slide 10.5- 19
CLASSROOM EXAMPLE 5 Rationalizing Binomial Denominators (cont’d)
Rationalize the denominator.
Solution:
Write radical quotients in lowest terms.
Objective 4
Slide 10.5- 20
Write the quotient in lowest terms.
24 36 7
16
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
Slide 10.5- 21
CLASSROOM EXAMPLE 6 Writing Radical Quotients in Lowest Terms
Solution:
Be careful to factor before writing a quotient in lowest terms.
22 32, 0
6
x xx
x
Factor the numerator.
Divide out common factors.
2 4 2
6
x x
x
2 1 2 2
6
x
x
1 2 22
2 3
x
x
1 2 2
3
Product rule
Slide 10.5- 22
CLASSROOM EXAMPLE 6 Writing Radical Quotients in Lowest Terms (cont’d)
Write the quotient in lowest terms.
Solution: