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