Root

The nth root of x: $\sqrt[n]{x} = x^{1/n}$.

WikipediaNth root↗MathWorldnthRoot↗WikidataQ601053↗Britannicatopic/root-mathematical-power↗

Root(x, n)the nth root $\sqrt[n]{x} = x^{1/n}$.

SageMathreal_nth_root(x, n)SymPyreal_root(x, n)Juliaenumeratio_root(x, n)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity) -> number	compute-engine	always

Domain: Arithmetic

Details

• The nth root: $\mathrm{Root}(x, n) = x^{1/n}$. See Sqrt for the special case $n = 2$.
• Odd roots of a negative number stay real: $\sqrt[3]{-8} = -2$. Even roots of a negative number canonicalize toward Sqrt instead, and evaluate exactly only when the radicand is itself a perfect power.
• A negative index gives the reciprocal root: $\mathrm{Root}(x, -n) = x^{-1/n}$.
• Both arguments are required -- unlike Sqrt, there's no default index.
• Evaluates exactly only when x is a perfect nth power; otherwise it stays in symbolic form, like $\sqrt[3]{2}$.

Plot

ComplexPlot

real slice

Examples

Basic #

The cube root of 27

live ↗

In$\sqrt[3]{27}$Out$3$

live ↗

In$\sqrt[4]{16}$Out$2$

Odd roots of a negative number stay real

live ↗

In$\sqrt[3]{-8}$Out$-2$

Stays symbolic when x isn't a perfect power

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In$\sqrt[3]{2}$Out$\sqrt[3]{2}$

A negative index gives the reciprocal root: $x^{-1/n}$

live ↗

In$\sqrt[-2]{16}$Out$\frac{1}{4}$

Scope #

The real cube root, Wolfram's $\mathrm{Surd}[-27, 3]$

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In$\sqrt[3]{-27}$Out$-3$

Any odd root of a negative number is real

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In$\sqrt[5]{-32}$Out$-2$

Approximate negative radicands stay real too

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In$\sqrt[5]{-3.5}$Out$-1.284\,735\,157\,123\,439\,338\,68$

Threads element-wise over a list

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In$\sqrt[3]{\bigl\lbrack-8, 27\bigr\rbrack}$Out$\bigl\lbrack-2, 3\bigr\rbrack$

Applications #

The edge length of a cube with volume 1000

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In$\sqrt[3]{1\,000}$Out$10$

Properties #

$(\sqrt[n]{x})^n = x$ for a perfect nth power

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In$\sqrt[3]{8}^3=8$Out$\mathrm{True}$

Root(x, 2) agrees with Sqrt

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In$\sqrt{16}=\sqrt{16}$Out$\mathrm{True}$

Possible issues #

An even root of a negative, non-perfect-power number canonicalizes to Sqrt and stays symbolic; numerically it's $2\sqrt{2}\,i$

live ↗

In$\sqrt{-8}$Out$\sqrt{-8}$

About

Examples	13
Checked against	Wolfram Language 8 agree5 differSageMath 12 agree1 errorSymPy 13 agreempmath 13 agreeJulia 13 agree
Crosswalk	WikipediaMathWorldWikidataBritannicaSageMathSymPyJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

An odd root of a negative real is the real one, as ours; mpmath's root is the principal root.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

See also: Sqrt, Square