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, $\sqrt[3]{x}={x}^{\frac{1}{3}}$.

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

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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 $\sqrt[3]{\frac{x}{y}}=\frac{\sqrt[3]{x}}{\sqrt[3]{y}}$.

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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 $\sqrt[3]{xy}\ne \sqrt[3]{x}\sqrt[3]{y}$.

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

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The cube root is a continuous mapping from $\mathbb{R}\to \mathbb{R}$.

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

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

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

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$\sqrt[3]{{x}^{3}{y}^{3}}=xy$ 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  20130322 11:57:22 
Last modified on  20130322 11:57:22 
Owner  Daume (40) 
Last modified by  Daume (40) 
Numerical id  12 
Author  Daume (40) 
Entry type  Definition 
Classification  msc 1100 
Related topic  NthRoot 
Related topic  SquareRoot 
Related topic  RationalNumber 
Related topic  IrrationalNumber 
Related topic  RealNumber 
Related topic  ComplexNumber 
Related topic  CubeOfANumber 