CubeRoot

The real cube root of x.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCubeRoot!↗

CubeRoot(x)the real cube root of x.

SageMathreal_nth_root(x, 3)SymPyreal_root(x, 3)Juliaenumeratio_cbrt(x)Rustcbrt(x)

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Elementary functions

Details

• Real-valued even at a negative x -- $\sqrt[3]{-27} = -3$, not a complex principal root. The same real branch as Root$(x, 3)$; CubeRoot is the named special case.
• Threads over a list, element-wise.
• An exact non-perfect-cube argument (like $\sqrt[3]{2}$) stays symbolic under plain evaluation; a floating-point argument, or N(), gives a decimal.
• A concretely complex argument is outside CubeRoot's real domain and is left unevaluated.

Plot

Examples

Basic #

The cube root of a perfect cube

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In$\mathrm{CubeRoot}(8)$Out$2$

The real root of a negative number, not a complex one

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In$\mathrm{CubeRoot}(-27)$Out$-3$

At a non-perfect cube, N() forces the numeric branch (plain evaluation stays exact and symbolic)

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In$N(\mathrm{CubeRoot}(2))$Out$1.259\,921\,049\,894\,873\,2$

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In$\operatorname{Series}(\mathrm{CubeRoot}(x), x, -1, 5)$Out$-1+\frac{x+1}{3}+\frac{(x+1)^2}{9}+\frac{1}{81}×5(x+1)^3+\frac{1}{243}×10(x+1)^4+\frac{1}{729}×22(x+1)^5+O\left((x+1)^6\right)$

Scope #

Threads element-wise over a list

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In$\mathrm{CubeRoot}(\bigl\lbrack-8, 27\bigr\rbrack)$Out$\bigl\lbrack-2, 3\bigr\rbrack$

Pulls out the perfect cube: $128 = 4^3\cdot2$

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In$\mathrm{CubeRoot}(128)$Out$4\sqrt[3]{2}$

A hundred digits of a real cube root

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In$N(\mathrm{CubeRoot}(18), 100)$Out$2.620\,741\,394\,208\,896\,607\,141\,661\,280\,441\,996\,270\,239\,427\,645\,723\,631\,725\,102\,773\,805\,728\,699\,819\,196\,042\,108\,828\,455\,825\,989\,073\,598$

Applied entrywise; negative fractions stay as unevaluated roots

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In$\mathrm{CubeRoot}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack-\frac{5}{3}, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\sqrt[3]{\frac{1}{2}}, -1\bigr\rbrack, \bigl\lbrack\sqrt[3]{-\frac{5}{3}}, \sqrt[3]{\frac{1}{2}}\bigr\rbrack\bigr\rbrack$

As a matrix function, the result involves nested radicals

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In$\mathrm{MatrixFunction}(\mathrm{CubeRoot}, \bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack-\frac{5}{3}, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}\sqrt[3]{1/2}\sqrt[3]{1-(2×15^{1/2})/3}+\frac{1}{2}\sqrt[3]{(1+(2×15^{1/2})/3)/2}, \frac{1}{10}\sqrt{15}\sqrt[3]{1/2}\sqrt[3]{1-(2×15^{1/2})/3}-\frac{1}{10}\sqrt{15}\sqrt[3]{(1+(2×15^{1/2})/3)/2}\bigr\rbrack, \bigl\lbrack\frac{1}{6}\sqrt{15}\sqrt[3]{1/2}\sqrt[3]{1-(2×15^{1/2})/3}-\frac{1}{6}\sqrt{15}\sqrt[3]{(1+(2×15^{1/2})/3)/2}, \frac{1}{2}\sqrt[3]{1/2}\sqrt[3]{1-(2×15^{1/2})/3}+\frac{1}{2}\sqrt[3]{(1+(2×15^{1/2})/3)/2}\bigr\rbrack\bigr\rbrack$

The cube root of 0 is 0

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In$\mathrm{CubeRoot}(0)$Out$0$

Infinite input stays infinite

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In$\mathrm{CubeRoot}(∞)$Out$∞$

Negative infinity is preserved, unlike a square root

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In$\mathrm{CubeRoot}(-∞)$Out$-∞$

Agrees with the real surd $\sqrt[3]{x}$

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In$\mathrm{CubeRoot}(x)=\mathrm{Surd}(x, 3)$Out$\mathrm{True}$

Differentiating the antiderivative recovers the integrand

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In$\mathrm{FullSimplify}(\frac{\mathrm{d}\int\!\mathrm{CubeRoot}(x)\, \mathrm{d}x}{\mathrm{d}x}=\mathrm{CubeRoot}(x))$Out$\mathrm{True}$

Stays unevaluated; no closed form for the definite integral

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In$\mathrm{FullSimplify}(\int_{0}^{5}\!\sin(x)\mathrm{CubeRoot}(x)\, \mathrm{d}x)$Out$\int_{0}^{5}\!\sin(x)\mathrm{CubeRoot}(x)\, \mathrm{d}x$

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In$\frac{\mathrm{d}\mathrm{CubeRoot}(x)}{\mathrm{d}x}$Out$\frac{1}{3x^{\frac{2}{3}}}$

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In$\mathrm{Solve}(\mathrm{CubeRoot}(x)=\frac{3}{2}, x)$Out$\bigl\lbrack\frac{27}{8}\bigr\rbrack$

The order-$k$ derivative is the falling-factorial power rule, $(1/3)^{\underline{k}}\, x^{-k} \sqrt[3]{x}$, real at a negative $x$ too

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In$\mathrm{FullSimplify}(\operatorname{D}(\mathrm{CubeRoot}(x), \bigl\lbrack x, k\bigr\rbrack))$Out$\frac{\mathrm{FallingFactorial}(\frac{1}{3}, k)\sqrt[3]{x}}{x^{k}}$

The coefficients about $x = 1$ are $\binom{1/3}{n}$; Wolfram writes the same sequence as a DifferenceRoot recurrence

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In$\mathrm{SeriesCoefficient}(\mathrm{CubeRoot}(x), \bigl\lbrack x, 1, n\bigr\rbrack)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack\binom{1/3}{n}, 0≤n\bigr\rbrack\bigr\rbrack, 0)$

The cube root is odd, so only the sine terms survive, with incomplete-gamma coefficients

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In$\mathrm{FullSimplify}(\mathrm{FourierSeries}(\mathrm{CubeRoot}(x), x, 1))$Out$\frac{2\sqrt{3}\sin(x)Γ(\frac{4}{3})-2i(\mathrm{ExpIntegralE}(-\frac{1}{3}, i π)-\mathrm{ExpIntegralE}(-\frac{1}{3}, -i π))\sin(x)π^{\frac{4}{3}}}{2π}$

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In$\mathrm{FullSimplify}(\int\!\mathrm{CubeRoot}(x)e^{x}\, \mathrm{d}x)$Out$\int\!\mathrm{CubeRoot}(x)e^{x}\, \mathrm{d}x$

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In$\operatorname{Series}(\mathrm{CubeRoot}(x), x, 1, 4)$Out$1+\frac{x-1}{3}-\frac{(x-1)^2}{9}+\frac{1}{81}×5(x-1)^3-\frac{1}{243}×10(x-1)^4+O\left((x-1)^5\right)$

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In$\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{CubeRoot}(x), x, \mathrm{x0}, 2))$Out$\frac{5\sqrt[3]{\mathrm{x0}}}{9}+\frac{5x}{9\mathrm{x0}^{\frac{2}{3}}}-\frac{x^2}{9\mathrm{x0}^{\frac{5}{3}}}+O\left((x-\mathrm{x0})^3\right)$

Properties #

The same as Root with index 3

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In$\mathrm{CubeRoot}(-64)=\sqrt[3]{-64}$Out$\mathrm{True}$

Unlike a square root, a negative real has a real cube root

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In$\mathrm{CubeRoot}(-8)$Out$-2$

About

Examples	25
Checked against	Wolfram Language 19 agree6 differSageMath 12 agree4 differ3 unevaluated1 errorSymPy 14 agree5 differ1 unevaluatedmpmath 12 agreeJulia 10 agree2 errorRust 9 agree
Crosswalk	Wolfram LanguageSageMathSymPyJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

The real cube root of a negative real, as ours; mpmath's cbrt is the principal root.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts