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


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


Objective 11

Find the least common denominator for a group of fractions.

Slide 7.3 - 3


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


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.


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


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.


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.


Objective 22

Rewrite rational expressions with given denominators.

Slide 7.3 - 9


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.)


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


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

chapter 7 section 3 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

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