College Algebra
Tutorial 11: Complex Rational Expressions
Learning Objectives
After completing this tutorial, you should be able to:
 Simplify complex fractions.

Introduction
Tutorial
A complex fraction is a rational expression that has a fraction
in its numerator, denominator or both.
In other words, there is at least one small fraction within the overall
fraction.
Some examples of complex fractions are:
and
There are two ways that you can simplify complex fractions.
We will call them method I and method
II. 
Method I
Simplifying a Complex Fraction

Combining only the numerator we get: 

*Rewrite fractions with LCD of ab

Combining only the denominator we get: 

*Rewrite fractions with LCD of a^2 b

Putting these back into the complex fraction we get: 

*Write numerator over denominator


*Rewrite div. as mult. of reciprocal
*Divide out a common factor of ab
*Excluded values of the original den. 
Note that the value that would be excluded from the domain is 0.
This is the value that makes the original denominator equal to 0. 
Combining only the numerator we get: 

*Rewrite fractions with LCD of (x  4)

Combining only the denominator we get: 

*Rewrite fractions with LCD of (x  4)

Putting these back into the complex fraction we get: 

*Write numerator over denominator


*Rewrite div. as mult. of reciprocal
*Divide out a common factor of (x  4)
*Excluded values of the original den. 
Note that the values that would be excluded from the domain are
4 and 16/5. These are the values that make the original denominators
equal to 0. 
Method II
Simplifying a Complex Fraction

The denominator of the numerator's fraction has the following factor: 
The denominator of the denominator's fraction has the following
factor: 
Putting all the different factors together and using the highest exponent,
we get the following LCD for all the small fractions: 
Multiplying numerator and denominator by the LCD we get: 

*Mult. num. and den. by (x + 1)(x  1)


*Divide out the common factor of x
*Excluded values of the original den. 
Note that the values that would be excluded from the domain are
1, 1, and 0. These are the values that make the original
denominator equal to 0. 
The denominators of the numerator's fractions have the following
factors: 
The denominators of the denominator's fractions have the following
factors: 
Putting all the different factors together and using the highest exponent,
we get the following LCD for all the small fractions: 
Multiplying numerator and denominator by the LCD we get: 

*Mult. num. and den. by (x + 5)(x  5)
*Excluded values of the original den. 
Note that the values that would be excluded from the domain are
5, 5, and 13/4. These are the values that make the original
denominators equal to 0. 
This rational expression cannot be simplified down any farther. 
Practice Problems
These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1b: Simplify.
Need Extra Help on these Topics?
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 15, 2009 by Kim Seward.
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