Beginning Algebra, 11th Edition

SECTION 7.3 Least Common Denominators 435

OBJECTIVES

Least Common Denominators

7.3

1 Find the least

common

denominator for a

group of fractions.

2 Write equivalent

rational expressions.

OBJECTIVE 1 Find the least common denominator for a group of frac-

tions. Adding or subtracting rational expressions often requires a least common

denominator (LCD).The LCD is the simplest expression that is divisible by all of

the denominators in all of the expressions. For example, the fractions

and have LCD 36,

because 36 is the least positive number divisible by both 9 and 12.

We can often find least common denominators by inspection. In other cases, we

find the LCD by a procedure similar to that used in Section 6.1for finding the great-

est common factor.

5

12

2

9

When each denominator is factored into prime factors, every prime factor must be a

factor of the least common denominator.

Finding the LCD

Find the LCD for each pair of fractions.

(a) (b)

Step 1 Write each denominator in factored form with numerical coefficients in

prime factored form.

Step 2 Find the LCD by taking each different factor the greatestnumber of times it

appears as a factor in any of the denominators.

15 = 3 # 5 10 x= 2 # 5 #x

24 = 2 # 2 # 2 # 3 = 23 # 3 8 x= 2 # 2 # 2 #x= 23 #x

1

8 x

,

3

10 x

1

24

,

7

15

EXAMPLE 1

The factor 2 appears three times in one

product and not at all in the other, so the

greatest number of times 2 appears is

three. The greatest number of times both

3 and 5 appear is one.

Step 3

Here, 2 appears three times in one product

and once in the other, so the greatest

number of times 2 appears is three. The

greatest number of times 5 appears is one,

and the greatest number of times x

appears in either product is one.

= 40 x NOW TRY

= 23 # 5 #x

LCD= 2 # 2 # 2 # 5 #x

= 120

= 23 # 3 # 5

LCD= 2 # 2 # 2 # 3 # 5

NOW TRY
EXERCISE 1
Find the LCD for each pair of
fractions.

(a) (b)

3

10 y

,

1

6 y

5

48

,

1

30

NOW TRY ANSWERS

1. (a) 240 (b) 30 y

Finding the Least Common Denominator (LCD)

Step 1 Factoreach denominator into prime factors.

Step 2 List each different denominator factorthe greatestnumber of

times it appears in any of the denominators.

Step 3 Multiplythe denominator factors from Step 2 to get the LCD.