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:
![\sqrt[3]{8} = 2](/img/math/268cbbda2dd912ea7e14c0527757ad39.png)
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
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,
and 
.
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
However for complex numbers we define instead the cube root to be
where ln(x) is the principal branch of the natural logarithm. If we write x as
- x = rexp(iθ)
where r is a non-negative real number and θ lies in the range
,
then the complex cube root is
![\sqrt[3]{x} = \sqrt[3]{r}\exp(i\theta/3).](/img/math/5ae75f33781d9a5467cbd7c822006fa7.png)
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.
See also
External links
- cube root on PlanetMath.