cube root

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cube root (dict)

In mathematics, the cube root (∛) of a number is a number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. For instance, the cube root of 8 is 2, because 2 × 2 × 2 = 8. This is written:

\sqrt 3 {8} = 2

Formally, the cube root of a real (or complex) number x is a real (correspondingly, complex) solution y to the equation:

y³ = x,

which leads to the equivalent notation for cube and other roots that

y = x^{1\over3}

A non-zero complex number has three cube roots. A real number has a unique real cube root, but when treated as a complex number it has two further cube roots, which are complex conjugates of each other. For instance, the cube roots of unity are 1,

-1 + \sqrt{3}i\over2

and

-1 - \sqrt{3}i\over2

. If R is one cube root of any real or complex number, the other two cube roots can be found by multiplying R by the two complex cube roots of unity. When treated purely as a real function of a real variable, we may define a real cube root for all real numbers by setting

(-x)^{1\over3} = -x^{1\over3}.

However for complex numbers we define instead the cube root to be

x^{1\over3} = \exp({\ln{x}\over3})

where

\ln{x}

is the principal branch of the natural logarithm. If we write x as

x = r \exp(i \theta)

where r is a non-negative real number and θ lies in the range

-\pi < \theta \le \pi

, then the complex cube root is

{r}\exp(i\theta/3).

This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. Hence, for instance,

\sqrt 3 {-8}

will not be -2, but rather

1+\sqrt{3}i

Cube Root

See also