Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8.3 - 1.

Copyright copy 2010 Pearson Education Inc All rights reserved Sec 83 - 1


Rational Expressions and Functions

Chapter 8


83

Complex Fractions


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



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



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



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





Simplify the numerator and

denominator (Step 1)


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)





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






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





Multiply the numerator

and denominator by

the LCD k(k ndash 2)


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



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




Both Methods


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


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


Simplify using Method 2

(Step 1)

Simplify (Step 2)


Rational Expressions and Functions

Chapter 8


83

Complex Fractions



Objectives




fractions




Complex Fractions



x

2x 7ndash

5mn

a2 ndash 16a + 2

a ndash 3

a2 + 4







denominator






(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)



Multiply

Simplify










3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)


83

Complex Fractions



Objectives




fractions




Complex Fractions



x

2x 7ndash

5mn

a2 ndash 16a + 2

a ndash 3

a2 + 4







denominator






(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)



Multiply

Simplify










3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)



Objectives




fractions




Complex Fractions



x

2x 7ndash

5mn

a2 ndash 16a + 2

a ndash 3

a2 + 4







denominator






(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)



Multiply

Simplify










3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)



Complex Fractions



x

2x 7ndash

5mn

a2 ndash 16a + 2

a ndash 3

a2 + 4







denominator






(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)



Multiply

Simplify










3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)







denominator






(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)



Multiply

Simplify










3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)





(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)



Multiply

Simplify










3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)








3x + 1x


(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


(Step 3)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)















= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)







= 1 (Step 1)


(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






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)






and denominator by




(b)k ndash 2

43k +

kk7ndash

=k ndash 2

43k + k(k ndash 2)


=k ndash 2



= 3k2(k ndash 2) + 4k







Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)




Both Methods


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




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)




Both Methods


Method 1 Method 2

(b)m1

n3ndash

mnn2 ndash 9m2

=mnn

mn3mndash

mnn2 ndash 9m2

=mn

n ndash 3mmn

n2 ndash 9m2divide

=mn


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 + 3m

1



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)



Negative Exponents





cb

bc

+

bc

cb

ndash bc

bc

c2 + b2

b2 ndash c2=

= bc

cb

ndash bcbc

cb

bc

+ bcbc


exponent



(Step 1)

Simplify (Step 2)

Date post:	17-Jan-2016
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