6.3 Simplifying Complex Fractions - DR. POLLONE'S WEBSITE

356 CHAPTER 6

6.3

Rational Expressions

Simplifying Complex Fractions

OBJECTIVES

1 Simplify Complex Fractions

by Simplifying the Numerator

and Denominator and Then

Dividing.

2 Simplify Complex Fractions

by Multiplying by a Common

Denominator.

3 Simplify Expressions with

A rational expression whose numerator, denominator, or both contain one or more

rational expressions is called a complex rational expression or a complex fraction.

Complex Fractions

x

1

1

x +

2y 2

y

a

b

6x - 2

y + 1

2

9y

The parts of a complex fraction are

x

f d Numerator of complex fraction

y + 2

Negative Exponents.

1

7 + f

y

d Main fraction bar

d Denominator of complex fraction

Our goal in this section is to simplify complex fractions. A complex fraction is simpliP

fied when it is in the form , where P and Q are polynomials that have no common

Q

factors. Two methods of simplifying complex fractions are introduced. The first method

evolves from the definition of a fraction as a quotient.

OBJECTIVE

1

Simplifying Complex Fractions: Method 1

Simplifying a Complex Fraction: Method I

Simplify the numerator and the denominator of the complex fraction so

that each is a single fraction.

Step 2. Perform the indicated division by multiplying the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction.

Step 3. Simplify if possible.

Step 1.

EXAMPLE 1

a.

Simplify each complex fraction.

2x

27y 2

5x

x + 2

b.

10

x - 2

6x2

9

1

x

+

y

y2

c.

y

1

+

2

x

x

Solution

a. The numerator of the complex fraction is already a single fraction, and so is the

2x

denominator. Perform the indicated division by multiplying the numerator,

, by

27y 2

6x2

. Then simplify.

the reciprocal of the denominator,

9

2x

27y 2

2x

6x2

=

,

9

27y 2

6x2

9

2x # 9

27y 2 6x2

2x # 9

=

27y 2 # 6x2

1

=

9xy 2

=

Multiply by the reciprocal of

6x 2

.

9

Section 6.3

Helpful Hint

Both the numerator and denominator are single fractions, so we

perform the indicated division.

5x

x + 2

5x

10

5x # x - 2

=

b.

,

=

x + 2

x - 2

x + 2

10

10

e

x - 2

5x1x - 22

= #

2 51x + 22

x1x - 22

=

21x + 22

Simplifying Complex Fractions 357

e

Multiply by the reciprocal of

10

.

x - 2

Simplify.

c. First simplify the numerator and the denominator of the complex fraction separately

so that each is a single fraction. Then perform the indicated division.

1#y

x

1

x

+

+ #

2

2

y

y y

y

y

=

y

y

1

1#x

+

+

x

x#x

x2

x2

x + y

=

=

=

=

y2

y + x

x2

x + y

y

2

Simplify the numerator. The LCD is y 2 .

Simplify the denominator. The LCD is x 2 .

Add.

#

x2

y + x

Multiply by the reciprocal of

y + x

x2

.

x2 1x + y2

y 2 1y + x2

x2

y2

Simplify.

PRACTICE

1

Simplify each complex fraction.

5k

36m

a.

15k

9

8x

x - 4

b.

3

x + 4

5

b

+ 2

a

a

c.

5a

1

+

2

b

b

CONCEPT CHECK

5

y

Which of the following are equivalent to ?

2

z

5

2

5#z

5

z

,

,

a.

c.

b.

y

y 2

y

z

2

OBJECTIVE

Answer to Concept Check:

a and b

2

Simplifying Complex Fractions: Method 2

Next we look at another method of simplifying complex fractions. With this method,

we multiply the numerator and the denominator of the complex fraction by the LCD

of all fractions in the complex fraction.

358 CHAPTER 6

Rational Expressions

Simplifying a Complex Fraction: Method 2

Multiply the numerator and the denominator of the complex fraction by

the LCD of the fractions in both the numerator and the denominator.

Step 2. Simplify.

Step 1.

EXAMPLE 2

Simplify each complex fraction.

x

1

+

2

y

y

b.

y

1

+

2

x

x

5x

x + 2

a.

10

x - 2

Solution

a. Notice we are reworking Example 1b using method 2. The least common denomina5x

10

5x

tor of

and

is 1x + 221x - 22 . Multiply both the numerator,

,

x + 2

x - 2

x + 2

10

and the denominator,

, by the LCD.

x - 2

5x

5x

a

b # 1x + 2 2 1x - 22

x + 2

x + 2

=

10

10

a

b # 1x + 22 1x - 22

x - 2

x - 2

Multiply numerator and

denominator by the LCD.

=

5 x # 1x - 22

2 # 5 # 1x + 22

Simplify.

=

x1x - 22

21x + 22

Simplify.

b. Here, we are reworking Example 1c using method 2. The least common denominax 1 y

1

tor of 2 , , 2 , and is x2y 2 .

x

y y x

x

1

x

1

+

a 2 + b # x 2y 2

2

y

y

y

y

=

y

y

1

1

+

a 2 + b # x 2y 2

x

x

x2

x

x # 2 2

1 # 2 2

x y +

x y

y

y2

=

y

# x2 y2 + 1 # x2 y2

2

x

x

3

2

x + xy

= 3

y + xy 2

x 21 x + y 2

= 2

y1y + x2

x2

= 2

y

PRACTICE

2

8x

x - 4

a.

3

x + 4

Use method 2 to simplify:

b

1

+

a

a2

b.

a

1

+

b

b2

Multiply the numerator and

denominator by the LCD.

Use the distributive property.

Simplify.

Factor.

Simplify.

Section 6.3

Simplifying Complex Fractions 359

OBJECTIVE

3

Simplifying Expressions with Negative Exponents

If an expression contains negative exponents, write the expression as an equivalent

expression with positive exponents.

EXAMPLE 3

Simplify.

x -1 + 2xy -1

x -2 - x -2y -1

Solution This fraction does not appear to be a complex fraction. If we write it by using

only positive exponents, however, we see that it is a complex fraction.

x -1 + 2xy -1

x -2 - x -2y -1

1

2x

+

y

x

=

1

1

- 2

x2

xy

1 2x 1

1

The LCD of , , 2 , and 2 is x2y. Multiply both the numerator and denominator

x y x

x

y

2

by x y.

a

=

2x # 2

1

+

b xy

y

x

1

1

- 2 b # x 2y

x2

xy

1# 2

2x # 2

xy +

xy

y

x

=

1 # 2

1

x y - 2 # x 2y

2

x

xy

x1y + 2x2 2

xy + 2x3

=

or

y - 1

y - 1

a

PRACTICE

3

Simplify:

EXAMPLE 4

Solution

Helpful Hint

Don¡¯t forget that 12x2

but 2x -1 = 2 #

1

2

= .

x

x

-1

1

,

=

2x

Simplify.

3x -1 + x -2y -1

y -2 + xy -1

Simplify:

12x2 -1 + 1

2x -1 - 1

1

+ 1

2x

=

2

- 1

- 1

x

(2x)-1 + 1

2x -1

Apply the

distributive

property.

Write using positive exponents.

1

+ 1b # 2x

2x

=

2

a - 1b # 2x

x

1 #

2x + 1 # 2x

2x

=

2#

2x - 1 # 2x

x

a

=

1 + 2x

4 - 2x

or

1 + 2x

212 - x2

The LDC of

1

2

and is 2x.

2x

x

Use distributive property.

Simplify.

360 CHAPTER 6

Rational Expressions

PRACTICE

4

Simplify:

13x2 -1 - 2

5x -1 + 2

Vocabulary, Readiness & Video Check

Complete the steps by writing the simplified complex fraction.

1.

7

x

z

1

+

x

x

=

7

xa b

x

z

1

xa b + xa b

x

x

=

2.

x

4

x2

1

+

2

4

=

x

4a b

4

x2

1

4a b + 4a b

2

4

=

Write with positive exponents.

3. x - 2 =

4. y -3 =

5. 2x -1 =

6. 12x2 -1 =

7. 19y2 -1 =

8. 9y -2 =

Martin-Gay Interactive Videos

Watch the section lecture video and answer the following questions.

OBJECTIVE

1

9. From Example 2, before you can rewrite the complex fraction as

division, describe how it must appear.

OBJECTIVE

2

10. How does finding an LCD in method 2, as in Example 3, differ from

finding an LCD in method 1? In your answer, mention the purpose of

the LCD in each method.

OBJECTIVE

3

See Video 6.3

6.3

11. Based on Example 4, what connection is there between negative

exponents and complex fractions?

Exercise Set

Simplify each complex fraction. See Examples 1 and 2.

10

3x

1.

5

6x

2

5

3.

3

2 +

5

4

x - 1

5.

x

x - 1

2

1 x

7.

4

x +

9x

2.

1 +

15

2x

5

6x

1

7

4

3 7

x

x + 2

2

x + 2

3

5 x

2

x +

3x

2 +

4.

6.

8.

4x 2 - y 2

xy

9.

2

1

y

x

x + 1

3

11.

2x - 1

6

2

3

+ 2

x

x

13.

4

9

x

x2

2

1

+ 2

x

x

15.

8

x + 2

x

10.

12.

14.

16.

x 2 - 9y 2

xy

1

3

y

x

x + 3

12

4x - 5

15

2

1

+

2

x

x

4

1

x

x2

1

3

+ 2

y

y

27

y + 2

y

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