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.



        91/2  =  3


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



        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.  



Which are real numbers:


  1. (1/3)1/4         Real, since the base is positive.

  2. (1/2)-1/6       Real, since the base is positive.

  3. (-4)1/8         Not real, since the base is negative and the exponent's denominator is even.

  4. (-5)-1/5        Real, since the exponent's denominator is odd.

We define xm/n by (x1/n)m  


xm/n  =  (x1/n)m  



        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



         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


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