Lesson Page

 

Rationalizing Denominators
 Math B

A fraction that contains a radical in its denominator can be written as an equivalent fraction with a rational denominator.

Never leave a radical in the denominator of a fraction.
Always rationalize the denominator.

1.  When the denominator is a monomial (one term), multiply both the numerator and the denominator by whatever makes the denominator an expression that can be simplified so that it no longer contains a radical.

* Sometimes the value being multiplied happens to be exactly the same as the denominator, as in Example 1:

  Example 1)        Simplify
 

Multiplying the top and bottom by will create the smallest perfect square under the square root in the denominator.
 
Replacing by 7 rationalizes the denominator.

 

* Sometimes you need to multiply by whatever makes the denominator a perfect square or perfect cube or any other power that can be simplified,
as in Examples 2 and 3.

  Example 2)         Simplify 
 

Multiply by a value that will create the smallest perfect square under the radical.  This will prevent the need for additional simplifications.

Choosing to multiply by (and not ) will create the smallest perfect square under the radical in the denominator.

 

                   

  Example 3)        Simplify 
 

Multiplying by will create the smallest perfect cube under the radical.

Replacing by 3, rationalizes the denominator.

 

Make sure you multiply by whatever makes the radicand
(the number under the radical sign) the smallest possible value to be simplified.  This will avoid having to further simplify later on.

                 

2.  When there is more than one term in the denominator, the process is a little tricky.  You will need to multiply the denominator by its conjugate.  The conjugate is the same expression as the denominator but with the opposite sign in the middle.

  Example 4)       Simplify

 

  Example 5)     Simplify

 

Be sure to enclose expressions with multiple terms in (  ). 
This will help you to remember to FOIL these expressions. 
Always reduce the root index (numbers outside radical) to simplest form (lowest) for the final answer. 

 

3.  When working with the reciprocal of a rational expression , if there is a radical in the denominator, you must rationalize the denominator.

  Example 6)      Write the reciprocal of


An expression is in simplest radial form if:

1. The radicand of an nth root has no nth powers in it,
    (radical part is fully reduced)

2. the root index is as low as possible,  and           
   (fraction outside radical is fully reduced)

3. there are no radicals in the denominator
   (rationalize the denominator).

 


DeMarr