WOLFRAM

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

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Basic Examples  (5)Summary of the most common use cases

CubeRoot gives a real root:

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Plot over a subset of the reals:

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Enter using cbrt:

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

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

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Series expansion:

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Scope  (36)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Evaluate to high precision:

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

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Evaluate efficiently at high precision:

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CubeRoot threads elementwise over lists and matrices:

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Compute average case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix CubeRoot function using MatrixFunction:

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Specific Values  (4)

Values of CubeRoot at fixed points:

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Values at zero:

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Values at infinity:

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Find a value of for which the using Solve:

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Substitute in the result:

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Visualize the result:

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Visualization  (3)

Plot the CubeRoot function:

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Visualize the absolute value and argument (sign) of :

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

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Polar plot with :

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Function Properties  (9)

CubeRoot is defined on the real numbers:

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The range of CubeRoot is all real numbers:

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

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is not an analytic function:

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Neither is it meromorphic:

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is non-decreasing:

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is injective:

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And surjective:

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is neither non-negative nor non-positive:

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is continuous on the reals but has a singularity at :

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It is singular because it is not differentiable:

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is neither convex nor concave:

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Differentiation  (3)

First derivative with respect to x:

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Higher derivatives with respect to x:

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Plot the higher derivatives with respect to x:

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Formula for the ^(th) derivative with respect to x:

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Integration  (4)

Compute the indefinite integral using Integrate:

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Verify the anti-derivative:

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Definite integral:

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Definite integral of CubeRoot over a symmetric interval is 0:

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More integrals:

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Series Expansions  (4)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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General term in the series expansion using SeriesCoefficient:

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The first-order Fourier series:

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Taylor expansion at a generic point:

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Function Identities and Simplifications  (3)

Primary definition:

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Products can be combined using FullSimplify:

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CubeRoot commutes with integer exponentiation:

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Applications  (1)Sample problems that can be solved with this function

Solve a differential equation with CubeRoot:

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Properties & Relations  (5)Properties of the function, and connections to other functions

CubeRoot is only defined for real inputs:

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CubeRoot is a bijection on the reals:

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Use CubeRoot to find real cube roots:

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

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The generating function for CubeRoot:

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Find the integral of a function containing CubeRoot:

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

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Possible Issues  (1)Common pitfalls and unexpected behavior

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

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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).

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 ]}

@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 ]}

@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 ]}