Rational (Fractional) Exponents
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Rational (fractional) exponents are an alternate way to express roots!

We're talking radicals here!

Notice:   The denominator of the rational exponent becomes the index of the radical,
and the numerator becomes the exponent of the radicand (expression inside the radical).

 

When you are dealing with a radical expression,
you can convert it to an expression
containing a rational (fractional) power.  This conversion may make the problem easier to solve.

Specifically,  ,    or in general:
 
   

 

We have already discussed simplifying radicals such as:
 

or

Let's look at these two problems in a new light!  When asked to simplify these radicals, it is often easier to rewrite the radicals using rational exponents and solve the problems by dealing with the laws of exponents.

Notice how applying the rules for dealing with the exponents makes quick work of the variables.

 

Look at these examples:

(1)   

(2)   

(3)   

 


When dealing with rational exponents, the Rules for Exponents are still valid!!!

Rule Example

 

Rationalizing radical denominators may often be accomplished more easily by using rational exponents.

Look at this example,
solved two ways.    

Simplify:  

Solved by Rationalizing the Denominator

Solved by Using Rational Exponents


Check out how these problems are done using rational exponents:

Evaluate: 

Evaluate: