Chapter 6 Section 5

Objectives

1

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Complex Fractions

Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2).

Simplify rational expressions with negative exponents.

6.5

2

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Complex Fractions.

The quotient of two mixed numbers can be written as a fraction.

Complex FractionA fraction with fractions in the numerator and/or denominator is called a complex fraction.

1 12 3

2 4

1 12 2

2 2 1 1

3 34 4

12

21

34

Numerator of complex fractionMain fraction barDenominator of complex fraction

Slide 6.5-3

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Objective 1

Simplify a complex fraction by multiplying numerator and denominator by the least common denominator. (Method 2 in our Text)

Slide 6.5-9

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Solution:

2 13 44 19 2

Simplify each complex fraction.

2 1

3 44 1

9

36

236

24 9

16 18

15

34

62

43

a

a

a

a

2 6

3 4

a

a

62

43

a

a

Slide 6.5-11

Simplifying Complex Fractions (Method 2)CLASSROOM EXAMPLE 4

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This technique utilizes the Fundamental Law of Fractions.

Method 2 from our Text

Simplifying a Complex Fraction

Step 1: Find the LCD of all fractions within the complex fraction.

Step 2: Multiply both the numerator and denominator of the complex fraction by this LCD using the distributive property as necessary. Write in lowest terms.

Slide 6.5-10

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Simplify the complex fraction.

2 2

2 2

2 3

4 1a b ab

a b ab

Solution:

2 2

22

2

2 2

2

2 3

4 1

a b ab

aa

bb

b

a

a

b

2 3

4

b a

ab

Slide 6.5-12

Simplifying a Complex Fraction (Method 2)CLASSROOM EXAMPLE 5

LCD = a2b2

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Solution:1 2

1 22 1

2 3

a b

b a

Simplify the complex fraction.

2 3

1 2 8

a b a

a a b

Slide 6.5-8

Simplifying a Complex Fraction CLASSROOM EXAMPLE 3

LCD = (a+1)(b-2)(a+3)

1 2

1 22 1

2 3

a b

b a

(a+1)(b-2)(a+3)

(a+1)(b-2)(a+3)

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Simplify each complex fraction.

1 21

41

x x

x

2

2

2 34

4 916

xxxx

2 34

2 3 2 3

4

44

4

44

x x

x x

xx

x x

x x

4

2 3

2 3

2 3

x

x

x

x

4

2 3

x

x

1 2 1

11

14

x xx

x

x

x x

1 2

4

x x

x

3 1

4

x

x

Slide 6.5-13

Deciding on a Method and Simplifying Complex Fractions

Solution:

CLASSROOM EXAMPLE 6

Note: One term distribute once, two terms distribute twice

Ex 1

Ex 2

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Simplify rational expressions with negative exponents.

Objective 2

Slide 6.5-10

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2 1

1 35

a b

a b

Simplify the expression, using only positive exponents in the answer.

2

3

1 1

1 5a b

a b

LCD = a2b3

22

33

3

2

1 1

1 5

a

a

bb

b

a

ba

2 3 2 3

2 3 2 3

2

3

1 1

1 5a b

a b

a b a b

a b a b

3 2 2

3 25

b a b

ab a

Slide 6.5-11

CLASSROOM EXAMPLE 7

Simplifying Rational Expressions with Negative Exponents

Solution:

Negative exponent means take the reciprocal of the base.

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3 1

3

2

2

x y

y x

Write with positive exponents.

3

3

1 2

2x yy x

LCD = x3y3

1

x y

Slide 6.5-12

CLASSROOM EXAMPLE 7 You Try It

Simplify the expression, using only positive exponents in the answer.

Solution:

3

3

3

3

1 2( )( )

( ( )) 2

x y

x yx yy x