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OBJECTIVES:

1. Simplify complex fractions by simplifying the numerator and denominator. (Method 1) 2. Simplify complex fractions by multiplying by a common denominator. (Method 2) 3. Compare the two methods of simplifying complex fractions. 4. Simplify rational expressions with negative exponents.

1. Simplify complex fractions by simplifying the numerator and denominator. (Method 1) 2. Simplify complex fractions by multiplying by a common denominator. (Method 2) 3. Compare the two methods of simplifying complex fractions. 4. Simplify rational expressions with negative exponents.

1.
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Sec 8.3 - 1

Sec 8.3 - 1

2.
Chapter 8

Rational Expressions and

Functions

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Sec 8.3 - 2

Rational Expressions and

Functions

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Sec 8.3 - 2

3.
8.3

Complex Fractions

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Sec 8.3 - 3

Complex Fractions

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Sec 8.3 - 3

4.
8.3 Complex Fractions

Objectives

1. Simplify complex fractions by simplifying the

numerator and denominator. (Method 1)

2. Simplify complex fractions by multiplying by a

common denominator. (Method 2)

3. Compare the two methods of simplifying complex

fractions.

4. Simplify rational expressions with negative

exponents.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 4

Objectives

1. Simplify complex fractions by simplifying the

numerator and denominator. (Method 1)

2. Simplify complex fractions by multiplying by a

common denominator. (Method 2)

3. Compare the two methods of simplifying complex

fractions.

4. Simplify rational expressions with negative

exponents.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 4

5.
8.3 Complex Fractions

Complex Fractions

A complex fraction is an expression having a fraction in the numerator,

denominator, or both. Examples of complex fractions include

2 – 7 a2 – 16

x 5 a+2

x , m , .

a–3

n

a2 + 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 5

Complex Fractions

A complex fraction is an expression having a fraction in the numerator,

denominator, or both. Examples of complex fractions include

2 – 7 a2 – 16

x 5 a+2

x , m , .

a–3

n

a2 + 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 5

6.
8.3 Complex Fractions

Simplifying Complex Fractions

Simplifying a Complex Fraction: Method 1

Step 1 Simplify the numerator and denominator separately.

Step 2 Divide by multiplying the numerator by the reciprocal of the

denominator.

Step 3 Simplify the resulting fraction, if possible.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 6

Simplifying Complex Fractions

Simplifying a Complex Fraction: Method 1

Step 1 Simplify the numerator and denominator separately.

Step 2 Divide by multiplying the numerator by the reciprocal of the

denominator.

Step 3 Simplify the resulting fraction, if possible.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 6

7.
8.3 Complex Fractions

EXAMPLE 1 Simplifying Complex Fractions by Method 1

Use Method 1 to simplify each complex fraction.

d+5

3d d+5 d–1

(a) = ÷ Write as a division problem.

d–1 3d 6d

6d

Both the numerator and the denominator are already simplified, so divide

by multiplying the numerator by the reciprocal of the denominator.

=

d+5

·

6d Multiply by the reciprocal of d – 1 .

3d d–1 6d

6d(d + 5) Multiply.

=

3d(d – 1)

2(d + 5)

= Simplify.

(d – 1)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 7

EXAMPLE 1 Simplifying Complex Fractions by Method 1

Use Method 1 to simplify each complex fraction.

d+5

3d d+5 d–1

(a) = ÷ Write as a division problem.

d–1 3d 6d

6d

Both the numerator and the denominator are already simplified, so divide

by multiplying the numerator by the reciprocal of the denominator.

=

d+5

·

6d Multiply by the reciprocal of d – 1 .

3d d–1 6d

6d(d + 5) Multiply.

=

3d(d – 1)

2(d + 5)

= Simplify.

(d – 1)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 7

8.
8.3 Complex Fractions

EXAMPLE 1 Simplifying Complex Fractions by Method 1

Use Method 1 to simplify each complex fraction.

2– 5 x

2x – 5

x x

2x – 5

x Simplify the numerator and

(b) = =

3 + 1x 3x

x +

1

x

3x + 1

x

denominator. (Step 1)

2x – 5 ÷ 3x + 1 Write as a division problem.

= x x

Multiply by the reciprocal of

2x – 5 x

= ·

x 3x + 1 3x + 1 . (Step 2)

x

2x – 5 Multiply and simplify.

=

3x + 1 (Step 3)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 8

EXAMPLE 1 Simplifying Complex Fractions by Method 1

Use Method 1 to simplify each complex fraction.

2– 5 x

2x – 5

x x

2x – 5

x Simplify the numerator and

(b) = =

3 + 1x 3x

x +

1

x

3x + 1

x

denominator. (Step 1)

2x – 5 ÷ 3x + 1 Write as a division problem.

= x x

Multiply by the reciprocal of

2x – 5 x

= ·

x 3x + 1 3x + 1 . (Step 2)

x

2x – 5 Multiply and simplify.

=

3x + 1 (Step 3)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 8

9.
8.3 Complex Fractions

Simplifying Complex Fractions

Simplifying a Complex Fraction: Method 2

Step 1 Multiply the numerator and denominator of the complex fraction by

the least common denominator of the fractions in the numerator

the fractions in the denominator of the complex fraction.

Step 2 Simplify the resulting fraction, if possible.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 9

Simplifying Complex Fractions

Simplifying a Complex Fraction: Method 2

Step 1 Multiply the numerator and denominator of the complex fraction by

the least common denominator of the fractions in the numerator

the fractions in the denominator of the complex fraction.

Step 2 Simplify the resulting fraction, if possible.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 9

10.
8.3 Complex Fractions

EXAMPLE 2 Simplifying Complex Fractions by Method 2

Use Method 2 to simplify each complex fraction.

2 – 5x 2 – 5x 2 – 5x · x Multiply the numerator and

(a) = ·1 =

3 + 1x 3+ x1 3 + 1x · x denominator by x since,

x

x = 1. (Step 1)

2·x – 5

x ·x

= Distributive property

3·x + 1

x ·x

2x – 5 Simplify. (Step 2)

=

3x + 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 10

EXAMPLE 2 Simplifying Complex Fractions by Method 2

Use Method 2 to simplify each complex fraction.

2 – 5x 2 – 5x 2 – 5x · x Multiply the numerator and

(a) = ·1 =

3 + 1x 3+ x1 3 + 1x · x denominator by x since,

x

x = 1. (Step 1)

2·x – 5

x ·x

= Distributive property

3·x + 1

x ·x

2x – 5 Simplify. (Step 2)

=

3x + 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 10

11.
8.3 Complex Fractions

EXAMPLE 2 Simplifying Complex Fractions by Method 2

Use Method 2 to simplify each complex fraction.

4 4 Multiply the numerator

3k + 3k + · k(k – 2)

k–2 k–2

(b) = and denominator by

k – 7 k – 7 · k(k – 2) the LCD, k(k – 2).

k k

4

3k [ k(k – 2) ] + · k(k – 2)

k–2

= Distributive property

k [ k(k – 2) ] – 7 · k(k – 2)

k

= 3k2(k – 2) + 4k

k2(k – 2) – 7(k – 2)

= 3k3 – 6k2 + 4k Multiply; lowest terms

k3 – 2k2 – 7k + 14

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 11

EXAMPLE 2 Simplifying Complex Fractions by Method 2

Use Method 2 to simplify each complex fraction.

4 4 Multiply the numerator

3k + 3k + · k(k – 2)

k–2 k–2

(b) = and denominator by

k – 7 k – 7 · k(k – 2) the LCD, k(k – 2).

k k

4

3k [ k(k – 2) ] + · k(k – 2)

k–2

= Distributive property

k [ k(k – 2) ] – 7 · k(k – 2)

k

= 3k2(k – 2) + 4k

k2(k – 2) – 7(k – 2)

= 3k3 – 6k2 + 4k Multiply; lowest terms

k3 – 2k2 – 7k + 14

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 11

12.
8.3 Complex Fractions

EXAMPLE 3 Simplifying Complex Fractions Using

Both Methods

Use both Method 1 and Method 2 to simplify each complex fraction.

Method 1 Method 2

3 3 3

x–1 x–1 x–1

(a) = (a)

4 4 4

x2 – 1 (x – 1)(x + 1) x2 – 1

3 4 3 · (x – 1)(x + 1)

= ÷

x–1 (x – 1)(x + 1) x–1

=

4 · (x – 1)(x + 1)

3 (x – 1)(x + 1) (x – 1)(x + 1)

= ·

x–1 4

3(x + 1) 3(x + 1)

= =

4 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 12

EXAMPLE 3 Simplifying Complex Fractions Using

Both Methods

Use both Method 1 and Method 2 to simplify each complex fraction.

Method 1 Method 2

3 3 3

x–1 x–1 x–1

(a) = (a)

4 4 4

x2 – 1 (x – 1)(x + 1) x2 – 1

3 4 3 · (x – 1)(x + 1)

= ÷

x–1 (x – 1)(x + 1) x–1

=

4 · (x – 1)(x + 1)

3 (x – 1)(x + 1) (x – 1)(x + 1)

= ·

x–1 4

3(x + 1) 3(x + 1)

= =

4 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 12

13.
8.3 Complex Fractions

EXAMPLE 3 Simplifying Complex Fractions Using

Both Methods

Use both Method 1 and Method 2 to simplify each complex fraction.

Method 1 Method 2

1 – 3 n – 3m 1 – 3 · mn

m n mn mn m n

(b) = (b)

n2 – 9m2 n2 – 9m2 n2 – 9m2 · mn

mn mn mn

1 mn – 3 mn

n – 3m 2

n – 9m 2

n

= ÷ =

m

mn mn

n2 – 9m2 · mn

mn

n – 3m mn

= · n – 3m n – 3m

mn (n – 3m)(n + 3m) = =

n2 – 9m2 (n – 3m)(n + 3m)

1 1

= =

n + 3m n + 3m

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 13

EXAMPLE 3 Simplifying Complex Fractions Using

Both Methods

Use both Method 1 and Method 2 to simplify each complex fraction.

Method 1 Method 2

1 – 3 n – 3m 1 – 3 · mn

m n mn mn m n

(b) = (b)

n2 – 9m2 n2 – 9m2 n2 – 9m2 · mn

mn mn mn

1 mn – 3 mn

n – 3m 2

n – 9m 2

n

= ÷ =

m

mn mn

n2 – 9m2 · mn

mn

n – 3m mn

= · n – 3m n – 3m

mn (n – 3m)(n + 3m) = =

n2 – 9m2 (n – 3m)(n + 3m)

1 1

= =

n + 3m n + 3m

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 13

14.
8.3 Complex Fractions

EXAMPLE 4 Simplifying Rational Expressions with

Negative Exponents

–1 –1

Simplify b c + bc , using only positive exponents in the answer.

bc –1 – b –1c

c b · bc Definition of negative

–1 –1

+ exponent

b c + bc =

b c

bc –1 –1

– b c b – c · bc Simplify using Method 2.

c b (Step 1)

c bc b bc

+

b c Distributive property

=

b bc – c bc

c b

= c2 + b2 Simplify. (Step 2)

b 2 – c2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 14

EXAMPLE 4 Simplifying Rational Expressions with

Negative Exponents

–1 –1

Simplify b c + bc , using only positive exponents in the answer.

bc –1 – b –1c

c b · bc Definition of negative

–1 –1

+ exponent

b c + bc =

b c

bc –1 –1

– b c b – c · bc Simplify using Method 2.

c b (Step 1)

c bc b bc

+

b c Distributive property

=

b bc – c bc

c b

= c2 + b2 Simplify. (Step 2)

b 2 – c2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.3 - 14