# Complex Fractions: Objectives and Examples

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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.
Sec 8.3 - 1
2. Chapter 8
Rational Expressions and
Functions
Sec 8.3 - 2
3. 8.3
Complex Fractions
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.
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
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.
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)
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)
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.
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
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
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
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
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