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

• 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. 1.

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

2. 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. 3.

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

4. 4.

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