Intermediate Algebra (11th edition)

Finding Least Common Denominators

Suppose that the given expressions are denominators of fractions. Find the LCD for

each group of denominators.

(a)

Each denominator is already factored.

Greatest exponent on xis 3.

Greatest exponent on yis 2.

(b) k

Each denominator is already factored. The LCD must be divisible by both and k.

LCD

It is usually best to leave a least common denominator in factored form.

(c)

Factor.

LCD

(d)

Factor.

LCD

(e)

Factor.

LCD= 21 m+ 321 m+ 2221 m- 12 NOW TRY

2 m^2 + 4 m- 6 = 21 m+ 321 m- 12

m^2 + 4 m+ 4 = 1 m+ 222

m^2 + 5 m+ 6 = 1 m+ 321 m+ 22

m^2 + 5 m+6, m^2 + 4 m+4, 2 m^2 + 4 m- 6

= 8 # 5 z# 1 z- 32 = 40 z 1 z- 32

5 z^2 - 15 z= 5 z 1 z- 32

8 z- 24 = 81 z- 32

8 z -24, 5 z^2 - 15 z

= 1 y- 421 y+ 221 y+ 12

y^2 + 3 y+ 2 = 1 y+ 221 y+ 12

y^2 - 2 y- 8 = 1 y- 421 y+ 22

y^2 - 2 y-8, y^2 + 3 y+ 2

= k 1 k- 32

k- 3

k-3,

= 10 x^3 y^2

LCD= 5 # 2 #x^3 #y^2

2 x^3 y= 2 #x^3 #y

5 xy^2 = 5 #x#y^2

5 xy^2 , 2 x^3 y

EXAMPLE 2

SECTION 7.2 Adding and Subtracting Rational Expressions 373

⎧

⎨

⎩

⎧

⎨

⎩

⎧

⎪

⎨

⎪

⎩

Don’t forget the factor k.

OBJECTIVE 3 Add and subtract rational expressions with different denom-

inators.First we must write each expression with the least common denominator by

multiplying its numerator and denominator by the factors needed to get the LCD. This is

valid because we are multiplying by a form of 1, the identity element for multiplication.

Consider the sum

The LCD for 15 and 12 is 60.

Fundamental property

Write each fraction with

the common denominator.

Add the numerators.

Keep the common denominator.

=

53

60

=

28 + 25

60

=

28

60

+

25

60

=

7 # 4

15 # 4

+

5 # 5

12 # 5

7

15

+

5

12

7

15 +

5

12.

NOW TRY
EXERCISE 2
Find the LCD for each group
of denominators.

(a)

(b)

(c)
x^2 + 10 x+ 25

3 x^2 + 9 x-30,x^2 - 4,

t, t- 8

15 m^3 n, 10 m^2 n

NOW TRY ANSWERS

2. (a) (b)
   (c) 31 x- 221 x+ 221 x+ 522

30 m^3 n t 1 t- 82