6.3 Simplifying Complex Fractions - DR. POLLONE'S WEBSITE

356 CHAPTER 6 Rational Expressions

6.3 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

Negative Exponents.

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

1

x

1

a

2y2

x+y

b 6x - 2 y + 1

2

9y

The parts of a complex fraction are

xf y+2

7

+

1 y

f

d Numerator of complex fraction 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

Step 1. 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.

EXAMPLE 1 2x 27y2

a. 6x2 9

Solution

Simplify each complex fraction.

5x b. x + 2

10 x-2

x1 y2 + y c. y 1 x2 + x

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

2x

denominator.

Perform

the

indicated division 6x2

by

multiplying

the

numerator,

27y2,

by

the reciprocal of the denominator, . Then simplify.

9

2x

27y2 6x2

=

2x 27y2

,

6x2 9

9

# 2x 9

= 27y2 6x2

6x2 Multiply by the reciprocal of .

9

=

2x # 9 # 27y2 6x2

=

1 9xy2

Section 6.3 Simplifying Complex Fractions 357

Helpful Hint

Both the numerator and denominator are single fractions, so we perform the indicated division.

e 5x

b.

x+2 e 10

=

5x x+2

,

10 x-2

=

5x # x - 2

x + 2 10

x-2

=

5x1x - 22

2 # 51x + 22

10

Multiply by the reciprocal of

.

x-2

=

x1x 21x

+

22 22

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.

x 1 x 1#y # y2 + y y2 + y y y 1 = y 1#x # x2 + x x2 + x x

x+y

y2 = y+x

x2

# =

x+ y2

y

x2 y+x

=

x2 1x y2 1y

+ +

y2 x2

=

x2 y2

Simplify the numerator. The LCD is y2. Simplify the denominator. The LCD is x2. Add.

y+x Multiply by the reciprocal of x2 .

Simplify.

PRACTICE

1 Simplify each complex fraction.

5k 36m a. 15k

9

8x b. x - 4

3 x+4

5b c. a + a2

5a 1 b2 + b

CONCEPT CHECK

5

y Which of the following are equivalent to ?

2

z

52 a. y , z

b. 5 # z

y2

5z c. y , 2

Answer to Concept Check: a and b

OBJECTIVE

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

Step 1. 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.

E X A M P L E 2 Simplify each complex fraction.

5x a. x + 2

10 x-2

Solution

x1 y2 + y b. y 1 x2 + x

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

tor

of

x

5x +

2

and

10 x-

2

is

1x

+

221x

-

22 .

Multiply

both

the

numerator,

x

5x +

, 2

10

and the denominator,

, by the LCD.

x-2

5x x+2

=

a

5x x+2

b#

1x

+

22 1x

-

22

10 x-2

a

x

10 -

2

b

#

1x

+

22

1x

-

22

=

5 x # 1x - 22 2 # 5 # 1x + 22

=

x1x 21x

+

22 22

Multiply numerator and denominator by the LCD.

Simplify. Simplify.

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

tor

of

x y2,

1 y,

y x2,

and

1 x

is

x2y2 .

x1 y2 + y

# x 1

a y2 + y b

x2y2

# y

= 1

x2 + x

y a x2

+

1 x

b

x2y2

# # x

y2

x2 y2

+

1 y

x2 y2

# # = y

x2

x2 y2 +

1 x

x2 y2

=

x3 y3

+ +

x2y xy2

=

x21 y21

x y

+ +

y 2 x 2

=

x2 y2

Multiply the numerator and denominator by the LCD.

Use the distributive property.

Simplify. Factor. Simplify.

PRACTICE

2 Use method 2 to simplify:

8x a. x - 4

3 x+4

b1 b. a2 + a

a1 b2 + b

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.

E X A M P L E 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.

1 2x

x-1 + 2xy-1 x-2 - x-2y-1

=

x+ 1

y 1

x2 - x2y

The LCD by x2y .

of

1x,

2yx,

1 x2,

and

1 x2y

is

x2y .

Multiply

both

the

numerator

and

denominator

PRACTICE

3

3x-1 + x-2y-1 Simplify: y-2 + xy-1

a

1 x

+

2x y

b

# x2y

=

# 1 1

a x2 - x2y b

x2y

1 # x2y

x

+

2x y

#

x2y

=

# # 1

x2

x2y

-

1 x2y

x2y

Apply the distributive property.

= xy + 2x3

or

x1y + 2x22 Simplify.

y-1

y-1

Helpful Hint

Don't

forget

that

12x2-1 =

1 ,

2x

# but 2x-1 = 2 1 = 2 .

xx

EXAMPLE 4

12x2-1 + 1 Simplify: 2x-1 - 1

Solution

1

(2x)-1 + 1 2x-1 - 1

=

2x + 1 2 x-1

Write using positive exponents.

=

a1 2x

+

1b # 2x

a

2 x

-

1b

# 2x

=

1 # 2x

2x

+

1 # 2x

2 x

#

2x

-

1

#

2x

= 1 + 2x 4 - 2x

or

1 + 2x 212 - x2

12 The LDC of and is 2x.

2x x Use distributive property. Simplify.

360 CHAPTER 6 Rational Expressions

PRACTICE

13x2-1 - 2

4

Simplify: 5x-1 + 2

Vocabulary, Readiness & Video Check

Complete the steps by writing the simplified complex fraction.

7 x

x

a

7 x

b

1.

=

=

1z x+x

x

a

1 x

b

+

x

a

z x

b

x

4axb

2.

4

x2 2+

1 4

=

4a x2 b 2

4

+

4a1b 4

=

Write with positive exponents. 3. x -2 =

4. y-3 =

5. 2x-1 = 7. 19y2-1 =

6. 12x2-1 = 8. 9y-2 =

Martin-Gay Interactive Videos See Video 6.3

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 11. Based on Example 4, what connection is there between negative exponents and complex fractions?

6.3 Exercise Set

Simplify each complex fraction. See Examples 1 and 2.

10 3x 1. 5 6x

2 1+5 3.

3 2+5

4 5. x - 1

x x-1

2 1-x 7.

4 x + 9x

15 2x 2. 5 6x

1 2+7 4.

4 3-7

x 6. x + 2

2 x+2

3 5-x 8.

2 x + 3x

4x2 - y2

xy 9.

21 y-x x+1

3 11.

2x - 1

6

23 13. x + x2

49 x2 - x

12 15. x + x2

8 x + x2

x2 - 9y2

xy 10.

13 y-x

x+3

12 12.

4x - 5

15

21 14. x2 + x

41 x2 - x

13 y + y2 16.

27 y + y2

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