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  (28)

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 x for which the CubeRoot[x]=3/2 using Solve:

Substitute in the result:

Visualize the result:

Visualization  (3)

Plot the CubeRoot[x] 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  (3)

CubeRoot is defined on the real numbers:

The range of CubeRoot is all real numbers:

Enter a character as \[CubeRoot], followed by a number:

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 ^(th) 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  (1)

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]:

Introduced in 2012
 (9.0)
 |
Updated in 2020
 (12.1)