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Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 2
Rational Expressions and Functions
Chapter 8
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 3
83
Complex Fractions
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 4
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 5
83 Complex Fractions
Complex Fractions
A complex fraction is an expression having a fraction in the numerator
denominator or both Examples of complex fractions include
x
2x 7ndash
5mn
a2 ndash 16a + 2
a ndash 3
a2 + 4
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 6
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 2
Rational Expressions and Functions
Chapter 8
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 3
83
Complex Fractions
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 4
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 5
83 Complex Fractions
Complex Fractions
A complex fraction is an expression having a fraction in the numerator
denominator or both Examples of complex fractions include
x
2x 7ndash
5mn
a2 ndash 16a + 2
a ndash 3
a2 + 4
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 6
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 3
83
Complex Fractions
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 4
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 5
83 Complex Fractions
Complex Fractions
A complex fraction is an expression having a fraction in the numerator
denominator or both Examples of complex fractions include
x
2x 7ndash
5mn
a2 ndash 16a + 2
a ndash 3
a2 + 4
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 6
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 4
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 5
83 Complex Fractions
Complex Fractions
A complex fraction is an expression having a fraction in the numerator
denominator or both Examples of complex fractions include
x
2x 7ndash
5mn
a2 ndash 16a + 2
a ndash 3
a2 + 4
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 6
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 5
83 Complex Fractions
Complex Fractions
A complex fraction is an expression having a fraction in the numerator
denominator or both Examples of complex fractions include
x
2x 7ndash
5mn
a2 ndash 16a + 2
a ndash 3
a2 + 4
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 6
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 6
83 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 copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 7
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
(a)
d + 53d
d ndash 16d
= d + 53d
d ndash 16d
divide
= d + 53d
6dd ndash 1
= 6d(d + 5)3d(d ndash 1)
= 2(d + 5)(d ndash 1)
Write as a division problem
Multiply by the reciprocal of d ndash 16d
Multiply
Simplify
Both the numerator and the denominator are already simplified so divide
by multiplying the numerator by the reciprocal of the denominator
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 8
83 Complex Fractions
EXAMPLE 1 Simplifying Complex Fractions by Method 1
Use Method 1 to simplify each complex fraction
Simplify the numerator and
denominator (Step 1)
Write as a division problem
3x + 1x
Multiply by the reciprocal of
(Step 2)
(b)2 x
5ndash
3 x1+
=x5ndashx
2x
x1+x
3x=
x2x ndash 5
x3x + 1
divide= x2x ndash 5
x3x + 1
=x
2x ndash 53x + 1
x
=3x + 12x ndash 5 Multiply and simplify
(Step 3)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 9
83 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 and
the fractions in the denominator of the complex fraction
Step 2 Simplify the resulting fraction if possible
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 10
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator and
denominator by x since
= 1 (Step 1)
Distributive property
(a)2 x
5ndash
3 x1+
Simplify (Step 2)
= 12 x
5ndash
3 x1+
= x2 x
5ndash
3 x1+ x
xx
=3 x x
1+ x
2 x xx5ndash
=3x + 12x ndash 5
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 11
83 Complex Fractions
EXAMPLE 2 Simplifying Complex Fractions by Method 2
Use Method 2 to simplify each complex fraction
Multiply the numerator
and denominator by
the LCD k(k ndash 2)
Distributive property
Multiply lowest terms
(b)k ndash 2
43k +
kk7ndash
=k ndash 2
43k + k(k ndash 2)
kk7ndash k(k ndash 2)
=k ndash 2
4+ k(k ndash 2) 3k [ k(k ndash 2) ]
k7ndash k(k ndash 2)k [ k(k ndash 2) ]
= 3k2(k ndash 2) + 4k
k2(k ndash 2) ndash 7(k ndash 2)
= 3k3 ndash 6k2 + 4k
k3 ndash 2k2 ndash 7k + 14
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 12
83 Complex Fractions
EXAMPLE 3 Simplifying Complex Fractions Using
Both Methods
Use both Method 1 and Method 2 to simplify each complex fraction
Method 1
(a)x ndash 1
3
x2 ndash 1
4=
x ndash 13
(x ndash 1)(x + 1)4
=x ndash 1
3(x ndash 1)(x + 1)
4divide
=x ndash 1
34
(x ndash 1)(x + 1)
=4
3(x + 1)
(a)x ndash 1
3
x2 ndash 1
4
=x ndash 1
3 (x ndash 1)(x + 1)
(x ndash 1)(x + 1)4 (x ndash 1)(x + 1)
=4
3(x + 1)
Method 2
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 13
83 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
(b)m1
n3ndash
mnn2 ndash 9m2
=mnn
mn3mndash
mnn2 ndash 9m2
=mn
n ndash 3mmn
n2 ndash 9m2divide
=mn
n ndash 3m(n ndash 3m)(n + 3m)
mn
=n + 3m
1
(b)m1
n3ndash
mnn2 ndash 9m2
mn
mn
=m1
n3ndash mnmn
mnn2 ndash 9m2
mn
=n ndash 3m
n2 ndash 9m2=
n ndash 3m
(n ndash 3m)(n + 3m)
=n + 3m
1
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)
Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 14
83 Complex FractionsEXAMPLE 4 Simplifying Rational Expressions with
Negative Exponents
Simplify using only positive exponents in the answerb ndash1c + bc ndash1
bc ndash1 ndash b ndash1c
b ndash1c + bc ndash1
bc ndash1 ndash b ndash1c=
cb
bc
+
bc
cb
ndash bc
bc
c2 + b2
b2 ndash c2=
= bc
cb
ndash bcbc
cb
bc
+ bcbc
Definition of negative
exponent
Distributive property
Simplify using Method 2
(Step 1)
Simplify (Step 2)