Definition

The cube root of a number is what you multiply by itself three times to get that number, written with a small $3$ inside the radical sign. Formally, the cube root of $n$ is the real number $x$ such that $x^3 = n$, written $\sqrt[3]{n}$ or $n^{1/3}$; unlike square roots, cube roots exist for negative numbers, since $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$, and every real number has exactly one real cube root. The function $x^{1/3}$ is continuously differentiable for $x \neq 0$, with $\frac{d}{dx}(x^{1/3}) = \frac{1}{3}x^{-2/3}$; in the complex plane, every non-zero $z$ has three cube roots, $|z|^{1/3} e^{i(\arg(z)+2\pi k)/3}$ for $k = 0, 1, 2$, forming the vertices of an equilateral triangle.