PlanetMath: cube root

PlanetMath

(more info)

Math for the people, by the people.

Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS

Login

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Owner confidence rating: Very low

Entry average rating: High

cube root

(Definition)

The cube root of a real number , written as $\sqrt[3]{x}$ , is the real number such that . Equivalently, $\sqrt[3]{x}^3 = x$ . Or, $\sqrt[3]{x}\sqrt[3]{x}\sqrt[3]{x} = x$ . The cube root notation is actually an alternative to exponentiation. That is, $\sqrt[3]{x} = x^\frac{1}{3}$ .

Properties:

The cube root operation of an exponentiation has the following property: $\sqrt[3]{x^n} = \sqrt[3]{x}^n$ .
The cube root operation is distributive for multiplication and division, but not for addition and subtraction. That is, $\sqrt[3]{xy} = \sqrt[3]{x} \sqrt[3]{y}$ , and $\sqrt[3]{\frac{x}{y}} = \frac{\sqrt[3]{x}}{\sqrt[3]{y}}$ .
However, in general, the cube root operation is not distributive for addition and substraction. That is, $\sqrt[3]{x + y} \not= \sqrt[3]{x} + \sqrt[3]{y}$ and $\sqrt[3]{x - y} \not= \sqrt[3]{x} - \sqrt[3]{y}$ .
The cube root is a special case of the general nth root.
The cube root is a continuous mapping from $\mathbb{R} \to \mathbb{R}$ .
The cube root function from $\mathbb{R} \to \mathbb{R}$ defined as $f(x)=\sqrt[3]{x}$ is an odd function.

Examples:

$\sqrt[3]{-8} = -2$ because $(-2)^3 = (-2) \times (-2) \times (-2) = -8$ .
$\sqrt[3]{x^3 + 3x^2 + 3x + 1} = x + 1$ because .
$\sqrt[3]{x^{3}y^{3}} = xy$ because $(xy)^3 = xy \times xy \times xy = x^{3}y^{3}$ .
$\sqrt[3]{\frac{8}{125}} = \frac{2}{5}$ because $(\frac{2}{5})^3 = \frac{2^3}{5^3} = \frac{8}{125}$ .

"cube root" is owned by Daume. [ full author list (2) | owner history (1) ]

(view preamble)

See Also: nth root, square root, rational number, irrational number, real number, complex number, cube of a number

Keywords:

cube root, root, arithmetic operator, operator

Log in to rate this entry.
(view current ratings)

Cross-references: odd function, function, continuous mapping, nth root, subtraction, addition, division, multiplication, distributive, property, operation, real number
There are 9 references to this entry.

This is version 8 of cube root, born on 2001-11-10, modified 2006-06-21.
Object id is 748, canonical name is CubeRoot.
Accessed 20042 times total.

Classification:

11-00 (Number theory :: General reference works )

Pending Errata and Addenda

None.[ View all 4 ]

Discussion

forum policy

Interact

post | correct | update request | add derivation | add example | add (any)