Transcript 10.5 Rationalizing Numerators and Denominators of Radical Expressions 1. Rationalize denominators. 2. Rationalize denominators that have a sum or difference with a square root term. 3.

10.5
Rationalizing Numerators and
Denominators of Radical Expressions
1. Rationalize denominators.
2. Rationalize denominators that have a sum or difference
with a square root term.
3. Rationalize numerators.
7 7 
49  7
5 5 
25  5
3 3 3
x x  x
7x  7x  7x
Rationalize the denominator
3
7
Irrational
number
7

7
3 7

7
Rational
number
Rationalize:
Factors or terms?
5 2
5
5

5
5 10

5

10
Rationalize:
1
4
9
16
25
36
49
64
81
100
6
12
6
12
3

3
6 3
6 3


36
6

3
1
6
12
6 12



12
12
12 2
12
2 3


2
2
3
3
6
6
3
3 3





3
12
2
3
3
3
1
3
3
Rationalize:
1
4
9
16
25
36
49
64
81
100
11

18
11
18
11
2


18
2
22
36

22
6
Rationalize:
1
4
9
16
25
36
49
64
81
100
11
7b

7b
7b
11
7b
11 7b

49b2
11 7b

7b
Rationalize:
1
8
27
64
125
216
343
3
3
3
6
5
6 3 25
3

5
25
3
150

3
125
3
150
5
5
5 25
3
2
75
150
Rationalize the denominator.
8
3
a) 2 3
3
3 2
b)
3
c) 2 6
3
3 6
d)
3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 9
Rationalize the denominator. 8
3
a) 2 3
3
3 2
b)
3
c) 2 6
3
3 6
d)
3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 10
Rationalize:
3
2 5
Can I still multiply by
52  5   2 5  5
Binomial Denominator
5 ?
No!
Can I square the denominator?
2 
5 2  5  4  4 5  5  9  4 5
No!
Try conjugates.
2 
5 2  5   4  5  1
To rationalize BINOMIAL denominators, use the conjugate!
Rationalize:
3
2 5
3
2  5
32  5 


2  5   2  5  4  2 5  2 5  25
Conjugates
32  5 

45
32  5 

1
 32  5 
 6  3 5
8
5 3
Rationalize:

8

5  3


2 10  2 6

2
All three!!!
5  3

5  3
Binomial Denominator
4∙10
40  24
25  15  15  9
5
3
2 10  2 6

2
2 10  6 

2

4∙6
10 
6
Factors or terms?
Rationalize:

5 3

2  4 6


5 3
24 6
2  4 6

2  4 6
Binomial Denominator
3 2
5 6  20 18
4  4 12  4 12  16 36
5 6  60 2

2  96
5 6  60 2

 94
 1 5 6  60 2 


1
 94
 5 6  60 2

94
Rationalize the numerator:
5
8
5
5
25
5



8 5
8 5
5
8
Rationalize the numerator:
2x  7

3


2x  7
3
2x  7

2x  7
4 x 2  7 2 x  7 2 x  49
3 2 x  21
2 x  49

3 2 x  21
Rationalize the denominator.
5
13  7
a) 5 13  5 7
6
b) 5 13  5 7
c)
6
13  7
5
d)
13  7
5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 17
Rationalize the denominator.
5
13  7
a) 5 13  5 7
6
b) 5 13  5 7
c)
6
13  7
5
d)
13  7
5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 18