CubeRoot
✖
CubeRoot
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 allBasic Examples (5)Summary of the most common use cases
CubeRoot gives a real root:

https://wolfram.com/xid/0b8cspipuq-5c7pw8

Plot over a subset of the reals:

https://wolfram.com/xid/0b8cspipuq-4wesvr


https://wolfram.com/xid/0b8cspipuq-bgm

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

https://wolfram.com/xid/0b8cspipuq-jdy82m

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

https://wolfram.com/xid/0b8cspipuq-vk17ac


https://wolfram.com/xid/0b8cspipuq-eeuq9z

Scope (36)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0b8cspipuq-wlv0g


https://wolfram.com/xid/0b8cspipuq-b0wt9

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

https://wolfram.com/xid/0b8cspipuq-y7k4a

Evaluate efficiently at high precision:

https://wolfram.com/xid/0b8cspipuq-di5gcr


https://wolfram.com/xid/0b8cspipuq-bq2c6r

CubeRoot threads elementwise over lists and matrices:

https://wolfram.com/xid/0b8cspipuq-ey0cqr


https://wolfram.com/xid/0b8cspipuq-erd7j

Compute average case statistical intervals using Around:

https://wolfram.com/xid/0b8cspipuq-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0b8cspipuq-thgd2

Or compute the matrix CubeRoot function using MatrixFunction:

https://wolfram.com/xid/0b8cspipuq-o5jpo

Specific Values (4)
Values of CubeRoot at fixed points:

https://wolfram.com/xid/0b8cspipuq-nww7l


https://wolfram.com/xid/0b8cspipuq-bmqd0y


https://wolfram.com/xid/0b8cspipuq-e5asej


https://wolfram.com/xid/0b8cspipuq-cnavla

Find a value of for which the
using Solve:

https://wolfram.com/xid/0b8cspipuq-bq8c92


https://wolfram.com/xid/0b8cspipuq-f2hrld


https://wolfram.com/xid/0b8cspipuq-3lx57

Visualization (3)
Plot the CubeRoot function:

https://wolfram.com/xid/0b8cspipuq-ecj8m7

Visualize the absolute value and argument (sign) of :

https://wolfram.com/xid/0b8cspipuq-51wwp7

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

https://wolfram.com/xid/0b8cspipuq-nn7oq4


https://wolfram.com/xid/0b8cspipuq-epb4bn

Function Properties (9)
CubeRoot is defined on the real numbers:

https://wolfram.com/xid/0b8cspipuq-cl7ele


https://wolfram.com/xid/0b8cspipuq-c8ej1c

The range of CubeRoot is all real numbers:

https://wolfram.com/xid/0b8cspipuq-evf2yr


https://wolfram.com/xid/0b8cspipuq-eaokrm

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

https://wolfram.com/xid/0b8cspipuq-44bmec


https://wolfram.com/xid/0b8cspipuq-h5x4l2


https://wolfram.com/xid/0b8cspipuq-e434t9


https://wolfram.com/xid/0b8cspipuq-g6kynf


https://wolfram.com/xid/0b8cspipuq-gi38d7


https://wolfram.com/xid/0b8cspipuq-ctca0g


https://wolfram.com/xid/0b8cspipuq-hkqec4


https://wolfram.com/xid/0b8cspipuq-hdm869

is neither non-negative nor non-positive:

https://wolfram.com/xid/0b8cspipuq-84dui

is continuous on the reals but has a singularity at
:

https://wolfram.com/xid/0b8cspipuq-xh65hl


https://wolfram.com/xid/0b8cspipuq-mdtl3h

It is singular because it is not differentiable:

https://wolfram.com/xid/0b8cspipuq-r7usb5

is neither convex nor concave:

https://wolfram.com/xid/0b8cspipuq-kdss3

Differentiation (3)
First derivative with respect to x:

https://wolfram.com/xid/0b8cspipuq-krpoah

Higher derivatives with respect to x:

https://wolfram.com/xid/0b8cspipuq-z33jv

Plot the higher derivatives with respect to x:

https://wolfram.com/xid/0b8cspipuq-fxwmfc

Formula for the derivative with respect to x:

https://wolfram.com/xid/0b8cspipuq-cb5zgj

Integration (4)
Compute the indefinite integral using Integrate:

https://wolfram.com/xid/0b8cspipuq-bponid


https://wolfram.com/xid/0b8cspipuq-q9uzaq

https://wolfram.com/xid/0b8cspipuq-op9yly


https://wolfram.com/xid/0b8cspipuq-bfdh5d

Definite integral of CubeRoot over a symmetric interval is 0:

https://wolfram.com/xid/0b8cspipuq-b9jw7l


https://wolfram.com/xid/0b8cspipuq-4nbst


https://wolfram.com/xid/0b8cspipuq-yncg8

Series Expansions (4)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0b8cspipuq-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0b8cspipuq-binhar

General term in the series expansion using SeriesCoefficient:

https://wolfram.com/xid/0b8cspipuq-dznx2j

The first-order Fourier series:

https://wolfram.com/xid/0b8cspipuq-f64drv

Taylor expansion at a generic point:

https://wolfram.com/xid/0b8cspipuq-jwxla7

Function Identities and Simplifications (3)

https://wolfram.com/xid/0b8cspipuq-d3qxpd

Products can be combined using FullSimplify:

https://wolfram.com/xid/0b8cspipuq-ewr40b

CubeRoot commutes with integer exponentiation:

https://wolfram.com/xid/0b8cspipuq-p2ubh6

Applications (1)Sample problems that can be solved with this function
Solve a differential equation with CubeRoot:

https://wolfram.com/xid/0b8cspipuq-hu0lfs


https://wolfram.com/xid/0b8cspipuq-dudq6m

Properties & Relations (5)Properties of the function, and connections to other functions
CubeRoot is only defined for real inputs:

https://wolfram.com/xid/0b8cspipuq-ivowrx

CubeRoot is a bijection on the reals:

https://wolfram.com/xid/0b8cspipuq-4tg1xd

Use CubeRoot to find real cube roots:

https://wolfram.com/xid/0b8cspipuq-xrr5bb

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

https://wolfram.com/xid/0b8cspipuq-j2zc6r

The generating function for CubeRoot:

https://wolfram.com/xid/0b8cspipuq-pz93yz


https://wolfram.com/xid/0b8cspipuq-dab3fj

Find the integral of a function containing CubeRoot:

https://wolfram.com/xid/0b8cspipuq-lxq9o

Visualize the function and the signed area between it and the axis:

https://wolfram.com/xid/0b8cspipuq-ytiy8

Wolfram Research (2012), CubeRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/CubeRoot.html (updated 2020).
Text
Wolfram Research (2012), CubeRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/CubeRoot.html (updated 2020).
Wolfram Research (2012), CubeRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/CubeRoot.html (updated 2020).
CMS
Wolfram Language. 2012. "CubeRoot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/CubeRoot.html.
Wolfram Language. 2012. "CubeRoot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/CubeRoot.html.
APA
Wolfram Language. (2012). CubeRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CubeRoot.html
Wolfram Language. (2012). CubeRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CubeRoot.html
BibTeX
@misc{reference.wolfram_2025_cuberoot, author="Wolfram Research", title="{CubeRoot}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CubeRoot.html}", note=[Accessed: 29-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_cuberoot, organization={Wolfram Research}, title={CubeRoot}, year={2020}, url={https://reference.wolfram.com/language/ref/CubeRoot.html}, note=[Accessed: 29-March-2025
]}