cube root

The cube root of a real number $x$, written as $\sqrt[3]{x}$, is the real number $y$ such that $y^3=x$. Equivalently, ${\sqrt[3]{x}}^3=x$. Or, $\sqrt[3]{x}\sqrt[3]{x}\sqrt[3]{x}=x$. The cube root notation is actually an alternative to exponentiation. That is, $\sqrt[3]{x}=x^{\frac{1}{3}}$.

Properties:

• The cube root operation of an exponentiation has the following property: $\sqrt[3]{x^n}={\sqrt[3]{x}}^n$.

• The cube root operation is distributive for multiplication and division, but not for addition and subtraction. That is, $\sqrt[3]{xy}=\sqrt[3]{x}\sqrt[3]{y}$, and $\sqrt[3]{\frac{x}{y}}=\frac{\sqrt[3]{x}}{\sqrt[3]{y}}$.

• However, in general, the cube root operation is not distributive for addition and substraction. That is, $\sqrt[3]{x+y}\ne \sqrt[3]{x}+\sqrt[3]{y}$ and $\sqrt[3]{x-y}\ne \sqrt[3]{x}-\sqrt[3]{y}$.

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

• The cube root is a continuous mapping from $ℝ\to ℝ$.

• The cube root function from $ℝ\to ℝ$ defined as $f(x)=\sqrt[3]{x}$ is an odd function.

Examples:

1. $\sqrt[3]{-8}=-2$ because ${(-2)}^3=(-2)\times (-2)\times (-2)=-8$.

2. $\sqrt[3]{x^3+3x^2+3x+1}=x+1$ because ${(x+1)}^3=(x+1)(x+1)(x+1)=(x^2+2x+1)(x+1)=x^3+3x^2+3x+1$.

3. $\sqrt[3]{x^3{y}^3}=xy$ because ${(xy)}^3=xy\times xy\times xy=x^3{y}^3$.

4. $\sqrt[3]{\frac{8}{125}}=\frac{2}{5}$ because ${(\frac{2}{5})}^3=\frac{2^3}{5^3}=\frac{8}{125}$.

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