6.3 Least Common Denominators

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.

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.

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 5.1 for finding the greatest common factor.

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.

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.

Slide 6.3-5

Find the least common denominator for a group of fractions. (cont’d)

Find the LCD for each pair of fractions.

Solution:

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

432 m 622 3 m

52 25

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

342 nm 2 53 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

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.

Slide 6.3-8

Finding LCDsCLASSROOM EXAMPLE 3

Objective 2

Write equivalent rational expressions.

Slide 6.3-9

Write equivalent rational expressions.

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 factor divided by itself. (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 ?

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

Slide 6.3-12

Writing Equivalent Rational ExpressionsCLASSROOM EXAMPLE 5