Rational Exponents

Square and Cube Roots as Exponents

We define a1/2 as the non-negative number such that when you square it, you get a.

Example

91/2  =  3

We define a1/3 as the number such that when you cube it you get a.

Example

81/3  =  2

Rational Roots

We define a1/n as the unique non-negative number x such that

xn  =  a

If n is odd then the domain of the function f(x) = x1/n is all real numbers and if n is even then the domain of f(x) = x1/n is all non-negative numbers.

Exercise

Which are real numbers:

1. (1/3)1/4

2. (1/2)-1/6

3. (-4)1/8

4. (-5)-1/5

We define xm/n by (x1/n)m

 xm/n  =  (x1/n)m

Example:

82/3  =   (81/3)2  =  22  =  4

In Radical notation the above can be written as:

Rules of Exponents

The same basic rules of exponents apply.  If you need a review of exponents, go to Rules of Exponents.  Or if you want to practice exponents interactively go to Practice Exponents

Example

x1/3 y -2/5
=  x1/3 + 2/3  y -2/5 - 1/4
x2/3 y1/4

=  x1y -8/20 - 5/20   =  x y -13/20