Cube root

In mathematics, a cube root of a number x is a number y such that y³ = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted ${\displaystyle {\sqrt[{3}]{8}}}$, is 2, because 2³ = 8, while the other cube roots of 8 are ${\displaystyle -1+i{\sqrt {3}}}$ and ${\displaystyle -1-i{\sqrt {3}}}$. The three cube roots of −27i are:

${\displaystyle 3i,\quad {\frac {3{\sqrt {3}}}{2}}-{\frac {3}{2}}i,\quad {\text{and}}\quad -{\frac {3{\sqrt {3}}}{2}}-{\frac {3}{2}}i.}$

In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign ${\displaystyle {\sqrt[{3}]{~^{~}}}.}$ The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always ${\displaystyle \left({\sqrt[{3}]{x}}\right)^{3}=x,}$ the cube of a nonzero number has more than one complex cube root and its principal cube root may not be the number that was cubed. For example, ${\displaystyle (-1+i{\sqrt {3}})^{3}=8}$, but ${\displaystyle {\sqrt[{3}]{8}}=2.}$