cube root

The cube root of a real number x, written as x3, is the real number y such that y3=x. Equivalently, x33=x. Or, x3x3x3=x. The cube root notation is actually an alternative to exponentiation. That is, x3=x13.


  • The cube root operation of an exponentiation has the following property: xn3=x3n.

  • The cube root operation is distributive for multiplication and division, but not for addition and subtractionPlanetmathPlanetmath. That is, xy3=x3y3, and xy3=x3y3.

  • However, in general, the cube root operation is not distributive for addition and substraction. That is, x+y3x3+y3 and x-y3x3-y3.

  • The cube root is a special case of the general nth root.

  • The cube root is a continuous mapping from .

  • The cube root function from defined as f(x)=x3 is an odd function.


  1. 1.

    -83=-2 because (-2)3=(-2)×(-2)×(-2)=-8.

  2. 2.

    x3+3x2+3x+13=x+1 because (x+1)3=(x+1)(x+1)(x+1)=(x2+2x+1)(x+1)=x3+3x2+3x+1.

  3. 3.

    x3y33=xy because (xy)3=xy×xy×xy=x3y3.

  4. 4.

    81253=25 because (25)3=2353=8125.

Title cube root
Canonical name CubeRoot
Date of creation 2013-03-22 11:57:22
Last modified on 2013-03-22 11:57:22
Owner Daume (40)
Last modified by Daume (40)
Numerical id 12
Author Daume (40)
Entry type Definition
Classification msc 11-00
Related topic NthRoot
Related topic SquareRoot
Related topic RationalNumber
Related topic IrrationalNumber
Related topic RealNumber
Related topic ComplexNumber
Related topic CubeOfANumber