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Lesson 22, Section 2

Complex fractions -- Division

Back to Section 1

  A complex fraction looks like this:    A complex fraction

The numerator and/or the denominator are themselves fractions.

To simplify a complex fraction, we can immediately apply the definition of division (Lesson 5):

a
b
  =  a·  1
b

Any fraction is equal to the numerator times the reciprocal
of the denominator.

Therefore,

divide fractions  =   p
q
·   n
m

Problem 1.   State in words how to simplify a complex fraction.

Rewrite it as the numerator times the reciprocal
of the denominator.

  Example 1.   Simplify    divide fractions
  Solution.    divide fractions   =   x2 − 25
   x8
·      x³ 
x − 5
    =   (x + 5)(x − 5)
        x8
·      x³ 
x − 5
 
    =   x + 5
   x5

Division -- which effectively this is -- becomes multiplication by the reciprocal.

divide fractions

on canceling the x + 2's.

Problem 2.   Simplify.

  a)     divide fractions   =    6 
x5
·   x2
8
  =     3  
4x3
  b)     divide fractions   =      4   
x − 1
·      1   
x − 1
  =         _4_      
x2 − 2x + 1
  c)     divide fractions   =   (x + 2) ·   x − 2
x + 2
  =   x − 2
  d)     divide fractions   =   x + 2
x + 1
·   x − 2
x − 1
  =   x2 − 4
x2 − 1
  e)     divide fractions   =       −h    
x(x + h)
·   1
h
  =   −       1     
x(x + h)

The h's cancel.  And according to the Rule of Signs, the product is negative.  (It's all right to leave the product in its factored form.)

  f)     divide fractions   =   (x + 2)(x − 2)
       3x2
·          _x_       
(x + 2)(x + 3)
  =   x − 2
  3x
·      1   
x + 3
  =      x − 2   
3x(x + 3)

Example 3.  If a complex fraction looks like this --

complex fraction

-- then we can simplify it by multiplying the numerator and denominator by c.

divide fractions

Problem 3.   Simplify the following.

divide fractions = _(x + 1) − 1_
(x + 1)(x − 1)
= __x__
x2 − 1
 
  Example 4.   Simplify    divide fractions

Solution.    1-over any number is its reciprocal.  Therefore,

divide fractions = 4
3

Problem 4.   Simplify the following.

  a)    divide fractions   =   x + 1
   x
  b)    divide fractions   =   x − 1
end

Next Lesson:  Adding algebraic fractions

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