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 The number of cube roots of a number depends on the number system that is considered.

Every nonzero real number x has exactly one real cube root that is denoted 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, neither a constructible number.

Every nonzero real of 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 and . The three cube roots of −27i are and 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 real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers, the real cube root is not the principal cube root. For positive real numbers, the principal cube root is the real cube root.

If y is any cube root of the complex number x, the other cube roots are and

In an algebraically closed field of characteristic different from three, every nonzero element has exactly three cube roots, which can be obtained from any of them by multiplying it by either root of the polynomial In an algebraically closed field of characteristic three, every element has exactly one cube root.

In other number systems or other algebraic structures, a number or element may have more than three cube roots. For example, in the quaternions, a real number has infinitely many cube roots.

Plot of y = 3x. The plot is symmetric with respect to origin, as it is an odd function. At x = 0 this graph has a vertical tangent.
A unit cube (side = 1) and a cube with twice the volume (side = 32 = 1.2599... OEISA002580).

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