Complex Fractions
Complex fractions have fractions in either the numerator, or
denominator, or usually both. These fractions can be simplified in one
of two ways, depending on the complex fraction.
If the numerator and/or the denominator of the complex fraction is a
fraction, remember that the fraction bar is just another way to indicate
division. We divide the fractions by multiplying by the reciprocal of the
second fraction (denominator of the complex fraction).
Example 1:
4x2
15y
8y3
5x5
4x2
8y3
15y
÷ 5x5
4x2
5x5
15y
· 8y3
x2 · x5
3 · y · 2 · y3
x2+5
3 · 2 · y1+3
Rewrite the complex fraction as a division
Multiply by the reciprocal of the second fraction
Reduce the coefficients by dividing out the
common factors 4 and 5
Use the product rule for exponents
x7
6y4
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 2:
m2 +m−6
m+4
m2 −9
m−3
m2 +m−6
m+4
m2 +m−6
m+4
÷
m2 −9
m−3
m−3
· m2 −9
Rewrite the complex fraction as division
Multiply by the reciprocal of the second fraction
(m+3)(m−2) ∙ (m−3)
(m+4) ∙ (m+3)(m−3)
m−2
Factor the polynomials
Divide out the common factors (m + 3) and (m – 3)
m+4
If the numerator and/or the denominator of the complex fraction contain
a fraction, the complex fraction can be rewritten as a simple fraction by
multiplying each term by the LCD of the fractions within the complex
fraction. As with any fraction, we will want to reduce the resulting
fraction if possible.
1
Example 3:
1− 2
x
1−
1
x
The LCD of the fractions with the complex fraction is x2
1
(x2 )1− 2(x2 )
x
1
x
(x2 )1− (x2 )
Multiply each term by the LCD
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
x2 −1
x2 −x
(x+1)(x−1)
x(x−1)
x+1
x
Simplified fraction
Factor the polynomials
Divide out the common factor (x + 1)
x−3 x+3
−
x+3 x−3
Example 4: x−3 x+3
+
x+3 x−3
The LCD of the fractions within the complex fraction is (x+3)(x-3)
x−3 x+3
(x+3)(x−3)
−
x+3 x−3
x−3 x+3
(x+3)(x−3)
(x+3)(x−3)
+
x+3 x−3
(x+3)(x−3)
Multiply each term by the LCD
(x−3)(x−3) − (x+3)(x+3)
(x−3)(x−3) + (x+3)(x+3)
Simplified fraction
(x2 −6x+9) − (x2 +6x+9)
(x2 −6x+9) + (x2 +6x+9)
Multiply
x2 −6x+9 − x2 − 6x−9
x2 −6𝑥+ 9 + x2 +6x+9
−12x
2x2 +18
−12x
2(x2 +9)
−6x
x2 +9
Distribute the negative
Combine like terms
Factor the polynomial
Divide out the common factor 2
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)