cbrt

cbrt denotes the cube root function, giving the real number y such that y^3 = x. In the real numbers, cbrt is defined for every real x, is strictly increasing, and serves as the inverse of the cubic function y^3. The notation ∛x or cbrt(x) is common, and for negative x the cube root is negative, for example cbrt(-8) = -2 and cbrt(0) = 0.

The real cube root has several key properties. It is an odd function, meaning cbrt(-x) = -cbrt(x). It maps the real line onto itself and grows like the cube root of x: as x → ∞, cbrt(x) ∼ x^(1/3). Differentiability holds for all x ≠ 0 with derivative cbrt'(x) = 1/(3 x^(2/3)); at x = 0 the derivative is not finite, so the function is not differentiable there in the usual sense.

When extended to the complex numbers, every nonzero x has three cube roots, related by multiplication by the cube roots of unity. Among these, the real cube root is the unique real root for real x; the other two roots are nonreal complex numbers.

In computing and programming, cbrt is a standard function name in many languages (for example in C’s math library as cbrt), returning the real cube root of a real input or handling real-valued arguments and sometimes offering variants for float or long double precision. Examples: cbrt(8) = 2; cbrt(-27) = -3.