# Cube root

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:

[itex]\sqrt[3]{8} = 2[itex]

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

[itex]y = x^{1\over3}[itex]

The cube root operation is associative with exponentiation and distributive with multiplication and division, but not addition and subtraction.

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 (1) are

1, [itex]-1 + \sqrt{3}i\over2[itex] and [itex]-1 - \sqrt{3}i\over2[itex].

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

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

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

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

where ln(x) is the principal branch of the natural logarithm. If we write x as

[itex]x = r \exp(i \theta)[itex]

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

[itex]-\pi < \theta \le \pi[itex],

then the complex cube root is

[itex]\sqrt[3]{x} = \sqrt[3]{r}\exp(i\theta/3).[itex]

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, ∛−8 will not be −2, but rather 1 + i√3.

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