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:

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

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 + √3i)/2 and (− 1 − √3i)/2 . If R is one cube root, 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 lnx 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 $-\pi < \phi \le \pi$ , then the complex cube root is

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

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