Cube root
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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:
- <math>\sqrt[3]{8} = 2<math>
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
- <math>y = x^{1\over3}<math>
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, <math>-1 + \sqrt{3}i\over2<math> and <math>-1 - \sqrt{3}i\over2<math>.
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
- <math>(-x)^{1\over3} = -x^{1\over3}.<math>
However for complex numbers we define instead the cube root to be
- <math>x^{1\over3} = \exp({\ln{x}\over3})<math>
where ln(x) is the principal branch of the natural logarithm. If we write x as
- <math>x = r \exp(i \theta)<math>
where r is a non-negative real number and θ lies in the range
- <math>-\pi < \theta \le \pi<math>,
then the complex cube root is
- <math>\sqrt[3]{x} = \sqrt[3]{r}\exp(i\theta/3).<math>
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.