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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 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, $\sqrt[3]{x+y}\not=\sqrt[3]{x}+\sqrt[3]{y}$ and $\sqrt[3]{xy}\not=\sqrt[3]{x}\sqrt[3]{y}$.

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

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

The cube root function from $\mathbb{R}\to\mathbb{R}$ 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 $(\frac{2}{5})^{3}=\frac{2^{3}}{5^{3}}=\frac{8}{125}$.
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Comments
Practical Examples?
Are there practical examples of the application of cube root other than as part of a larger computation?
Re: Practical Examples?
Suppose that you know the volume of a cubeformed cistern or other, for example 6 cubic units (e.g. cubic meters or cubic feet).
If you then want to know what are the measures of this cube, i.e. what is the length of the edge of cube, then you must to take the cube root of the volume; thus the length of the edge is about 1.817 corresponding length units (e.g. meters or feet).
Cheers,
Jussi