Chapter 7 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
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Transcript of Chapter 7 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Chapter Chapter 77Section Section 33
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Least Common Denominators
Find the least common denominator for a group of fractions.Rewrite rational expressions with given denominators.
11
22
7.37.37.37.3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Find the least common denominator for a group of fractions.
Slide 7.3 - 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find the least common denominator for a group of fractions.
Adding or subtracting rational expressions often requires
a least common denominator (LCD), the simplest
expression that is divisible by all of the denominators in all
of the expressions. For example, the least common
denominator for the fractions and is 36, because 36 is
the smallest positive number divisible by both 9 and 12.
Slide 7.3 - 4
2
9
5
12
We can often find least common denominators by
inspection. For example, the LCD for and is 6m. In
other cases, we find the LCD by a procedure similar to that
used in Section 6.1 for finding the greatest common factor.
1
62
3m
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find the least common denominator for a group of fractions. (cont’d)
To find the least common denominator, use the following steps.
Step 1: Factor each denominator into prime factors.
Slide 7.3 - 5
Step 2: List each different denominator factor the greatest number of times it appears
in any of the denominators.
Step 3: Multiply the denominator factors from Step 2 to get the LCD.
When each denominator is factored into prime factors, every prime factor must be a factor of the least common denominator.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Find the LCD for each pair of fractions.
Solution:
Finding the LCD
Slide 7.3 - 6
7 1,
10 25
4 6
4 11,
8 12m m
10 2 5
4 48 2 2 2m m
25 5 5
6 612 2 2 3m m
432 m 622 3 m
52 25
2LCD 5 2
3 63D 2LC m
50
624m
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find the LCD for
EXAMPLE 2 Finding the LCD
Slide 7.3 - 7
Solution:
3 5
4 5 and .
16 9m n m
3 316 2 2 2 2m n m n 5 59 3 3m m
342 nm 2 53 m
4 2 52 3LCD m n 5144m nWhen finding the LCD, use each factor the greatest number of times it appears in any single denominator, not the total number of times it appears.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
4 1,
1 1x x
Finding the LCD
Slide 7.3 - 8
Find the LCD for the fractions in each list.
2 2
6 3 1,
4 16
x
x x x
4x x
4 4x x
LCD 4 4x x x
4x x
44 xx
Either x − 1 or 1 − x, since they are opposite expressions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Rewrite rational expressions with given denominators.
Slide 7.3 - 9
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Rewrite rational expressions with given denominators.
Once the LCD has been found, the next step in preparing to add or subtract two rational expressions is to use the fundamental property to write equivalent rational expressions.
Step 1: Factor both denominators.
Slide 7.3 - 10
Step 2: Decide what factor(s) the denominator must be multiplied by in order to equal the
specified denominator.Step 3: Multiply the rational expression by the factor
divided by itself. (That is, multiply by 1.)
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Rewrite each rational expression with the indicated denominator.
EXAMPLE 4
Solution:
3
4
93
4 9
3
4 9
?
4
242
30
k
k
3 ?
4 36
7 ?
5 65
k
k
7 7
5 5
6
6
k k k
k
Writing Rational Expressions with Given Denominotors
Slide 7.3 - 11
7 ?
5 30
k
k
27
36
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Rewrite each rational expression with the indicated denominator. Solution:9 ?
2 5 6 15a a
Writing Rational Expressions with Given Denominators
Slide 7.3 - 12
2
5 1 ?
2 2 1
k
k k k k k
9 ?
32 5 2 5a a
9 9
2 5 2
3
35a a
27
6 15a
1
5 1 ?
2 2
k
k k kk k
5 1 5 1 1
2 12
k k
k k k k
k
k
5 1 1
2 1
k k
k k k