(b)

=

xy

y-x

=

y+x

xy

x

(^2) y 2

1 y-x 21 y+ x 2

=

y+x

xy

,

y^2 - x^2

x^2 y^2

y+x

xy

y^2 - x^2

x^2 y^2

=

y

xy

+

x

xy

y^2

x^2 y^2

-

x^2

x^2 y^2

=

1

x

+

1

y

1

x^2

-

1

y^2

SECTION 7.3 Complex Fractions 383

(b)

= NOW TRY

xy

y-x

=

xy 1 y+x 2

1 y+x 21 y- x 2

=

x y^2 +x^2 y

y^2 - x^2

a

1

x

b x^2 y^2 +a

1

y

b x^2 y^2

a

1

x^2

b x^2 y^2 - a^

1

y^2

b x^2 y^2

=

a

1

x

+

1

y

b #x^2 y^2

a

1

x^2

-

1

y^2

b#x^2 y^2

=

1

x

+

1

y

1

x^2

-

1

y^2

OBJECTIVE 4 Simplify rational expressions with negative exponents. To

simplify, we begin by rewriting the expressions with only positive exponents.

Simplifying Rational Expressions with Negative Exponents

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

(a) (Section 5.1)

Write with positive exponents.

Distributive property

= Lowest terms

mp^2 +m^2

2 p^2 - m^2 p

m^2 p^2 #

1

m

+m^2 p^2 #

1

p^2

m^2 p^2 #

2

m^2

• m^2 p^2 #

1

p

=

m^2 p^2 a

1

m

+

1

p^2

b

m^2 p^2 a

2

m^2

-

1

p

b

=

1

m

+

1

p^2

2

m^2

-

1

p

=

a-n=a^1 n

m-^1 + p-^2

2 m-^2 - p-^1

EXAMPLE 4

Simplify by Method 2, multiplying

the numerator and denominator by

the LCD, m^2 p^2.

2 m-^2 = 2 #m-^2 =^21 #m^1 2 =m^2 2

The base of 2m

is m, not 2m:

2 m-^2 =m^22.

• 2

NOW TRY
EXERCISE 3
Simplify each complex
fraction by both methods.

(a) (b)

1

m^2

• 1
  n^2
  1
  m

• 1
  n

1

p- 6

5

p^2 - 36

NOW TRY ANSWERS

3. (a) (b)nmn-m
   p+ 6
   5