# CubeRoot

CubeRoot[x]

gives the real-valued cube root of x.

# Details • CubeRoot[x] returns the real-valued cube root for real-valued x.
• For symbolic x in CubeRoot[x], x is assumed to be real valued.
• CubeRoot can be evaluated to arbitrary numerical precision.
• CubeRoot automatically threads over lists.
• In StandardForm, CubeRoot[x] formats as .
• can be entered as cbrt .
• ∛z can also be used for input. The character is entered as cbrti or \[CubeRoot].

# Examples

open allclose all

## Basic Examples(5)

CubeRoot gives a real root:

Plot over a subset of the reals:

Enter using cbrt :

Note that this is not the same as , which is Power[x,1/3]:

Compare the real and imaginary parts of and over the reals:

Series expansion:

## Scope(34)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

CubeRoot threads elementwise over lists and matrices:

### Specific Values(4)

Values of CubeRoot at fixed points:

Values at zero:

Values at infinity:

Find a value of for which the using Solve:

Substitute in the result:

Visualize the result:

### Visualization(3)

Plot the CubeRoot function:

Visualize the absolute value and argument (sign) of :

The function has the same absolute value but a different argument for :

Polar plot with :

### Function Properties(9)

CubeRoot is defined on the real numbers:

The range of CubeRoot is all real numbers:

Enter a character as \[CubeRoot], followed by a number: is not an analytic function:

Neither is it meromorphic: is non-decreasing: is injective:

And surjective: is neither non-negative nor non-positive: is continuous on the reals but has a singularity at :

It is singular because it is not differentiable: is neither convex nor concave:

### Differentiation(3)

First derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x:

Formula for the  derivative with respect to x:

### Integration(4)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Definite integral of CubeRoot over a symmetric interval is 0:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

The first-order Fourier series:

Taylor expansion at a generic point:

### Function Identities and Simplifications(3)

Primary definition:

Products can be combined using FullSimplify:

CubeRoot commutes with integer exponentiation:

## Properties & Relations(4)

CubeRoot is only defined for real inputs:

CubeRoot is a bijection on the reals:

Use CubeRoot to find real cube roots:

Use Power[x,1/3] or to find the principal complex cube root:

The generating function for CubeRoot:

## Possible Issues(1)

On the negative real axis, CubeRoot[x] is different from the principal root returned by Power[x,1/3]: