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Chapter 7. Section 3. Complex Fractions. Simplify complex fractions by simplifying the numerator and denominator (Method 1). Simplify complex fractions by multiplying by a common denominator (Method 2). Compare the two methods of simplifying complex fractions. - PowerPoint PPT Presentation
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Section 3Chapter 7
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objectives
2
3
4
Complex Fractions
Simplify complex fractions by simplifying the numerator and denominator (Method 1).
Simplify complex fractions by multiplying by a common denominator (Method 2).
Compare the two methods of simplifying complex fractions.
Simplify rational expressions with negative exponents.
7.3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
A complex fraction is a quotient having a fraction in the numerator, denominator, or both.
2
2
1, , and
41 9
.2
13 3
61
myx m
my m
Slide 7.3- 3
Complex Fractions
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify complex fractions by simplifying the numerator and denominator (Method 1).
Objective 1
Slide 7.3- 4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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.
Slide 7.3- 5
Simplify complex fractions by simplifying the numerator and denominator (Method 1).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use Method 1 to simplify the complex fraction.
2
23
yyyy
Both the numerator and denominator are already simplified.
3
2 2y
y
y
y
2 3
2
y y
y y
3( 2)
2
y
y
Write as a division problem.
Multiply by the reciprocal.
Multiply.
Slide 7.3- 6
CLASSROOM EXAMPLE 1
Simplifying Complex Fractions (Method 1)
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
34
15
x
x
Simplify the numerator and denominator.
4 3
5 1
xx xxx x
34
15
xxxxx x
4 3
5 1
xxxx
4 3 5 1x x
x x
4 3
5 1
x x
x x
4 3
5 1
x
x
Slide 7.3- 7
CLASSROOM EXAMPLE 1
Simplifying Complex Fractions (Method 1) (cont’d)
Use Method 1 to simplify the complex fraction.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify complex fractions by multiplying by a common denominator (Method 2).
Objective 2
Slide 7.3- 8
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 and the fractions in the denominator of the complex fraction.
Step 2 Simplify the resulting fraction if possible.
Slide 7.3- 9
Simplify complex fractions by multiplying by a common denominator (Method 2).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use Method 2 to simplify the complex fraction.
34
15
x
x
The LCD is x. Multiply the numerator and denominator by x.
34
15
xx
xx
4 3
5 1
x
x
34
15
x x
x
x
xx
Slide 7.3- 10
CLASSROOM EXAMPLE 2
Simplifying Complex Fractions (Method 2)
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
43
13
2
yy
yy
Multiply the numerator and denominator by the LCD y(y + 1).
( 1)4
31
32 ( 1)
y y
y
yy
yy
y
[ ( 1)] ( 1)
[ ( 1)]
43
1
2 ( )3
1
y y y y
y
y
y y
y
yy y
2
2
3 ( 1) 4
2 ( 1) 3( 1)
y y y
y y y
3 2
3 2
3 3 4
2 2 3 3
y y y
y y y
Slide 7.3- 11
CLASSROOM EXAMPLE 2
Simplifying Complex Fractions (Method 2) (cont’d)
Use Method 2 to simplify the complex fraction.
Solution:
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Compare the two methods of simplifying complex fractions.
Objective 3
Slide 7.3- 12
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2
52
34
y
y
Simplify the complex fraction by both methods.
Method 1 Method 2
52
3( 2)( 2)
y
y y
5 3
2 ( 2)( 2)y y y
( 2)(
2
5 2)
3
y
y
y
5( 2)
3
y
52
3( 2)( 2)
y
y y
( 2)( 2)
( 2)( 2)
52
3( 2)( 2)
y
yy
y
y y
y
5( 2)
3
y
Slide 7.3- 13
CLASSROOM EXAMPLE 3
Simplifying Complex Fractions (Both Methods)
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2 2
1 1
1 1a b
a b
Method 1
2 2
2 2 2 2
b aab abb aa b a b
LCD = ab
LCD = a2b2
2 2
2 2
b aab
b aa b
2 2
( )( )
b aab
b a b aa b
2 2
( )( )b a b a b a
ab a b
2 2
( )( )
a b
ab b
b a
aa b
ab
b a
Slide 7.3- 14
CLASSROOM EXAMPLE 3
Simplifying Complex Fractions (Both Methods) (cont’d)
Solution:
Simplify the complex fraction by both methods.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2 2
1 1
1 1a b
a b
Method 2
LCD of the numerator and denominator is a2b2.
2 2
2 2
ab a b
b a
( )
( )( )
ab b a
b a b a
ab
b a
2
2
2
2 22
1 1
1 1a b
a b
a b
a b
Slide 7.3- 15
CLASSROOM EXAMPLE 3
Simplifying Complex Fractions (Both Methods) (cont’d)
Solution:
Simplify the complex fraction by both methods.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify rational expressions with negative exponents.
Objective 4
Slide 7.3- 16
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 7.3- 17
CLASSROOM EXAMPLE 4
Simplifying Rational Expressions with Negative Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
3 1
3
2
2
x y
y x
Write with positive exponents.
3
3
1 2
2x yy x
LCD = x3y3 3
3
31 2
2
y xy xx yy x
3
3
3
2
21
y xx yy x
3 3
3
2 2
1
y x y x
x y
3
3 3
2 1
2
y x
x y y x
3
1
x y
Slide 7.3- 18
CLASSROOM EXAMPLE 4
Simplifying Rational Expressions with Negative Exponents (cont’d)
Simplify the expression, using only positive exponents in the answer.
Solution:
