fraction power

Let m be an integer and n a positive factor of m.  If x is a positive real number, we may write the identical equation


and therefore the definition of nth root ( gives the

xmn=xmn. (1)

Here, the exponentMathworldPlanetmath mn is an integer.  For enabling the validity of (1) for the cases where n does not divide m we must set the following

Definition.  Let mn  be a fractional number, i.e. an integer m not divisible by the integer n, which latter we assume to be positive.  For any positive real number x we define the  fraction powerxmn as the nth

xmn:=xmn. (2)


  1. 1.

    The existence of the in the right hand side of (2) is proved here (

  2. 2.

    The defining equation (2) is independent on the form of the exponent mn:  If  kl=mn,  then we have  (xmn)ln=[(xmn)n]l=xlm=xkn=[(xkl)l]n=(xkl)ln,  and because the mapping  yyln  is injective in +, the positive numbers xkl and xmn must be equal.

  3. 3.

    The fraction power functionxxmn  is a special case of power functionDlmfDlmf.

  4. 4.

    The presumption that x is positive signifies that one can not identify all nth roots ( xn and the powers x1n.  For example, -83 equals -2 and  26=13,  but one must not

    (-8)13=(-8)26=(-8)26=646= 2.

    The point is that (-8)13 is not defined in .  Here we have  l=6  and the mapping  yyln  is not injective in  -+.  — Nevertheless, some people and books may use also for negative x the equality  xn=x1n  and more generally  xmn=xmn  where one then insists that  gcd(m,n)=1.

  5. 5.

    According to the preceding item, for the negative values of x the derivative of odd roots (, e.g. x3, ought to be calculated as follows:


    The result is similar as dx3dx for positive x’s, although the root functions are not special cases of the power function.

Title fraction power
Canonical name FractionPower
Date of creation 2014-09-21 12:12:39
Last modified on 2014-09-21 12:12:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 21
Author pahio (2872)
Entry type Definition
Classification msc 26A03
Synonym fractional power
Related topic PowerFunction
Related topic GeneralPower
Related topic IntegrationOfFractionPowerExpressions
Related topic NthRootFormulas