When we simplify radicals with exponents, we divide the exponent by the index. Another way to write division is with a fraction bar. This idea is how we will define rational exponents.
The denominator of a rational exponent becomes the index on our radical. Likewise, the index on the radical becomes the denominator of the exponent. We can use this property to change any radical expression into an exponential expression.
Index is denominator 

Negative exponents from reciprocals 
We can also change any rational exponent into a radical expression by using the denominator as the index.
Index is denominator 

Negative exponent means reciprocals 
The ability to change between exponential expressions and radical expressions allows us to evaluate problems we had no means of evaluating before by changing to a radical.

Change to radical, denominator is index, negative means reciprocal  

Evaluate radical  

Evaluate exponent  

Our Solution 
The largest advantage of being able to change a radical expression into an exponential expression is we are now allowed to use all our exponent properties to simplify. The following table reviews all of our exponent properties.


When adding and subtracting with fractions, we need to be sure to have a common denominator. When multiplying, we only need to multiply the numerators together and denominators together. The following examples show several different problems, using different properties to simplify rational exponents.
EXAMPLE

Need common denominator on and  

Add exponents on and  

Our Solution 
EXAMPLE

Multiply each exponent by  

Our Solution 
EXAMPLE

In numerator, need common denominator to add exponents 


Subtract exponents on x, reduce exponents on y 


Negative exponent moves down to denominator 


Our Solution 
EXAMPLE

Need common denominator on x's in parentheses; use 


Subtract exponents 


Multiply by  

Our Solution 