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 all close allBasic Examples (5)
CubeRoot gives a real root:
CubeRoot[-27]Plot over a subset of the reals:
Plot[CubeRoot[x], {x, -2, 2}]
:Surd[-4.5, 3]Note that this is not the same as 
, which is Power[x,1/3]:
-4.5^(1/(3))Compare the real and imaginary parts of 
and 
over the reals:
ReImPlot[{x^(1/(3)), Surd[x, 3]}, {x, -4, 4}, PlotLegends -> Automatic]Series[CubeRoot[x], {x, -1, 5}]Scope (36)
Numerical Evaluation (6)
CubeRoot[128.]N[CubeRoot[18], 100]The precision of the output tracks the precision of the input:
CubeRoot[0.2111111555555555555111111111111111]Evaluate efficiently at high precision:
CubeRoot[12`100]//TimingCubeRoot[181`1000000];//TimingCubeRoot threads elementwise over lists and matrices:
CubeRoot[{0, 1.5, 8, Infinity}]CubeRoot[(| | |
| :- | :---- |
| 1 | u |
| v | 3.375 |)]//MatrixFormCompute average case statistical intervals using Around:
CubeRoot[ Around[2 / 3, 0.001]]Compute the elementwise values of an array:
CubeRoot[{{1 / 2, -1}, {-5 / 3, 1 / 2}}]Or compute the matrix CubeRoot function using MatrixFunction:
MatrixFunction[CubeRoot, {{1 / 2, -1}, {-5 / 3, 1 / 2}}]Specific Values (4)
Values of CubeRoot at fixed points:
Table[CubeRoot[n ], {n, {-27, -8, -1, 1, 8, 27}}]CubeRoot[0]CubeRoot[Infinity]CubeRoot[-Infinity]Find a value of 
for which the 
using Solve:
Solve[CubeRoot[x] == 3 / 2]xval = x /. First[%]Plot[CubeRoot[x], {x, -10, 10}, Epilog -> Style[Point[{xval, CubeRoot[xval]}], PointSize[Large], Red]]Visualization (3)
Plot the CubeRoot function:
Plot[CubeRoot[x], {x, -3, 3}]Visualize the absolute value and argument (sign) of 
:
AbsArgPlot[Surd[x, 3], {x, -4, 4}, PlotLegends -> Automatic]The function 
has the same absolute value but a different argument for 
:
AbsArgPlot[x^(1/(3)), {x, -4, 4}, PlotLegends -> Automatic]
PolarPlot[CubeRoot[ϕ], {ϕ, 0, 8π}, Frame -> True]Function Properties (9)
CubeRoot is defined on the real numbers:
FunctionDomain[CubeRoot[x], x]FunctionDomain[CubeRoot[x], x, Complexes]The range of CubeRoot is all real numbers:
FunctionRange[CubeRoot[x], x, y]FunctionRange[CubeRoot[x], x, y, Complexes]//ReduceEnter a ∛ character as \[CubeRoot], followed by a number:
∛-1000
FunctionAnalytic[Surd[``x``, 3], x]
FunctionMeromorphic[Surd[``x``, 3], x]
FunctionMonotonicity[Surd[``x``, 3], x]
FunctionInjective[Surd[``x``, 3], x]Plot[{Surd[``x``, 3], 1}, {x, -5, 5}]FunctionSurjective[Surd[``x``, 3], x]Plot[{Surd[``x``, 3], -2}, {x, -10, 5}]
is neither non-negative nor non-positive:
FunctionSign[Surd[``x``, 3], x]
is continuous on the reals but has a singularity at 
:
FunctionContinuous[Surd[``x``, 3], x]FunctionSingularities[Surd[``x``, 3], x]It is singular because it is not differentiable:
Limit[(CubeRoot[h] - CubeRoot[0]/h), h -> 0]
is neither convex nor concave:
FunctionConvexity[Surd[``x``, 3], x]Differentiation (3)
First derivative with respect to x:
D[CubeRoot[x], x]Higher derivatives with respect to x:
Table[D[CubeRoot[x], {x, k}], {k, 1, 3}]//FullSimplifyPlot the higher derivatives with respect to x:
Plot[%, {x, -7, 7}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]Formula for the 
derivative with respect to x:
D[CubeRoot[x], {x, k}]// FullSimplifyIntegration (4)
Compute the indefinite integral using Integrate:
Integrate[CubeRoot[x], x]FullSimplify[D[Integrate[CubeRoot[x], x], x] == CubeRoot[x]]FullSimplify[D[%, x] == CubeRoot[x]]Integrate[CubeRoot[x], {x, 0, 4}]Definite integral of CubeRoot over a symmetric interval is 0:
Integrate[CubeRoot[x], {x, -4, 4}]Integrate[Exp[x]CubeRoot[x], x]//FullSimplifyIntegrate[Sin[x]CubeRoot[x], {x, 0, 5}]//FullSimplifySeries Expansions (4)
Find the Taylor expansion using Series:
Series[CubeRoot[x], {x, 1, 4}]Plots of the first three approximations around 
:
terms = Normal@Table[Series[CubeRoot[x], {x, 1, m}], {m, 1, 5, 2}];
Plot[{CubeRoot[x], terms}, {x, -1, 3}]General term in the series expansion using SeriesCoefficient:
SeriesCoefficient[CubeRoot[x], {x, 1, n}]The first-order Fourier series:
FourierSeries[CubeRoot[x], x, 1]// FullSimplifyTaylor expansion at a generic point:
Series[CubeRoot[x], {x, x0, 2}]// FullSimplifyFunction Identities and Simplifications (3)
CubeRoot[x] == Surd[x, 3]Products can be combined using FullSimplify:
CubeRoot[x]CubeRoot[y] == CubeRoot[x y]CubeRoot commutes with integer exponentiation:
Reduce[Table[CubeRoot[x ^ n] == CubeRoot[x] ^ n, {n, 10}], Reals]Applications (1)
Solve a differential equation with CubeRoot:
DSolve[3 x y'[x] - y[x] == 0, y[x], x]DSolve[{CubeRoot[y[x]] - y'[x] == 0, y[0] == 1}, y[x], x]//QuietProperties & Relations (5)
CubeRoot is only defined for real inputs:
FunctionDomain[CubeRoot[x], x, ℂ]CubeRoot is a bijection on the reals:
FunctionBijective[CubeRoot[x], x]Use CubeRoot to find real cube roots:
CubeRoot[-8.]Use Power[x,1/3] or 
to find the principal complex cube root:
-8.^(1/(3))The generating function for CubeRoot:
GeneratingFunction[CubeRoot[n], n, x]Series[%, {x, 0, 5}]Find the integral of a function containing CubeRoot:
Integrate[(1 + x^3) CubeRoot[x], {x, -1, 1}]Visualize the function and the signed area between it and the 
axis:
Plot[(1 + x^3) CubeRoot[x], {x, -1, 1}, Filling -> Axis, FillingStyle -> {RGBColor[0.880722, 0.611041, 0.142051, 0.5], RGBColor[0.368417, 0.506779, 0.709798, 0.5]}]Related Links
MathWorldText
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.
APA
Wolfram Language. (2012). CubeRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CubeRoot.html