Cube root

In mathematics, a cube root of a number x is a number y that has the given number as its third power; that is y 3 = x . {\displaystyle y^{3}=x.} The number of cube roots of a number depends on the number system that is considered. Every real number x has exactly one real cube root that is denoted x 3 {\textstyle {\sqrt[{3}]{x}}} and called the real cube root of x or simply the cube root of x in contexts where complex numbers are not considered. For example, the real cube roots of 8 and −8 are respectively 2 and −2. The real cube root of an integer or of a rational number is generally not a rational number, nor a constructible number. Every nonzero real or complex number has exactly three cube roots that are complex numbers. If the number is real, one of the cube roots is real and the two other are nonreal complex conjugate numbers. Otherwise, the three cube roots are all nonreal. For example, the real cube root of 8 is 2 and the other cube roots of 8 are − 1 + i 3 {\displaystyle -1+i{\sqrt {3}}} and − 1 − i 3 {\displaystyle -1-i{\sqrt {3}}} . The three cube roots of −27i are 3 i , 3 3 2 − 3 2 i , {\displaystyle 3i,{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i,} and − 3 3 2 − 3 2 i . {\displaystyle -{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i.} The number zero has a unique cube root, which is zero itself. The cube root is a multivalued function. The principal cube root is its principal value, that is a unique cube root that has been chosen once for all. The principal cube root is the cube root with the largest real part. In the case of negative...