Chapter 6 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Least Common...

Chapter 6 Section 3

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Least Common Denominators

Find the least common denominator for a group of fractions.

Write equivalent rational expressions.

6.3

2


Objective 1

Find the least common denominator for a group of fractions.

Slide 6.3-3


Find the least common denominator for a group of fractions

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 number divisible by both 9 and 12.

2

9

5

12

We can often find least common denominator by inspection.

For example, the LCD for and is 6m.

What about these denominators makes it easy to see the LCD?

1

62

3m

Slide 6.3-4


Finding the Least Common Denominator (LCD)

Step 1: Factor each denominator into prime factors.

Step 2: List each different denominator factor the greatest number of times it appears in any of the denominators.

MAKE LCD SOUP

Step 3: Multiply the selected denominator factors to obtain the LCD

When each denominator is factored into prime factors, every prime factor must be a factor in the least common denominator soup.

Do not put too many factors in the soup!

Slide 6.3-5


Find the LCD for each pair of fractions.

Solution: Factor Make Soup

7 1,

10 25

4 6

9 11,

8 12m m

10 2 5

4 48 2 2 2m m

25 5 5

6 612 2 2 3m m

2LCD 5 2

3 63D 2LC m

50

624m

Slide 6.3-6

Finding the LCDCLASSROOM EXAMPLE 1


Find the LCD for

Solution:

3 5

4 5 and .

16 9m n m

3 316 2 2 2 2m n m n 5 59 3 3m m

4 2 52 3LCD m n 5144m n

When 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.

Slide 6.3-7

Finding the LCDCLASSROOM EXAMPLE 2


Solution:

4 1,

1 1x x

Find the LCD for the fractions in each list.

2 2

6 3 1,

4 16

x

x x x

2 4x x2 16x

LCD 4 4x x x

4x x

44 xx

Either x − 1 or 1 − x, since they are opposite expressions.

Slide 6.3-8

Finding LCDsCLASSROOM EXAMPLE 3


Objective 2

Write equivalent rational expressions.

Slide 6.3-9


Writing A Rational Expression with a Specified Denominator

Step 1: Factor both denominators.

Slide 6.3-10

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 same amount, numerator and denominator. (That is, multiply by 1.)


Rewrite each rational expression with the indicated denominator.

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 7 ?

5 30

k

k

27

36

Slide 6.3-11

Writing Equivalent Rational ExpressionsCLASSROOM EXAMPLE 4


Rewrite each rational expression with the indicated denominator.

Solution:

9 ?

2 5 6 15a a

2

5 1 ?

2 2 1

k

k k k k k

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

Slide 6.3-12

Writing Equivalent Rational ExpressionsCLASSROOM EXAMPLE 5

Chapter 6 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Least Common...

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