gives the real-valued cube root of x.
- 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].
Examplesopen 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:
Numerical Evaluation (4)
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:
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:
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:
Compute the indefinite integral using Integrate:
Verify the anti-derivative:
Definite integral of CubeRoot over a symmetric interval is 0:
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)
Products can be combined using FullSimplify:
CubeRoot commutes with integer exponentiation:
Properties & Relations (1)
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
Updated in 2020