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cube root (Definition)

The cube root of a real number $ x$, written as $ \sqrt[3]{x}$, is the real number $ y$ such that $ y^3 = x$. 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:

Examples:

  1. $ \sqrt[3]{-8} = -2$ because $ (-2)^3 = (-2) \times (-2) \times (-2) = -8$.
  2. $ \sqrt[3]{x^3 + 3x^2 + 3x + 1} = x + 1$ because $ (x + 1)^3 = (x + 1)(x + 1)(x + 1) = (x^2 + 2x + 1)(x + 1) = x^3 + 3x^2 + 3x + 1$.
  3. $ \sqrt[3]{x^{3}y^{3}} = xy$ because $ (xy)^3 = xy \times xy \times xy = x^{3}y^{3}$.
  4. $ \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) ]
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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
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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:
AMS MSC11-00 (Number theory :: General reference works )

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Practical Examples? by SoftTest123 on 2007-08-13 08:01:41
Are there practical examples of the application of cube root other than as part of a larger computation?
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