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 number (or complex number) 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}

The cube root operation is associativity with exponentiation and distributivity with multiplication and division (mathematics), 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,

-1 + \sqrt{3}i\over2

and

-1 - \sqrt{3}i\over2

. 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 :

(-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 ln(''x'') is the principal branch of the natural logarithm. If we write x as :

x = r \exp(i \theta)

where ''r'' is a non-negative real number and θ lies in the range :

-\pi < \theta \le \pi

, then the complex cube root is :

\sqrt[3]{x} = \sqrt[3]{r}\exp(i\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, ∛−8 will not be −2, but rather 1 + i√3. ==See also== * Radical (mathematics) * Square root * Root of unity * Shifting nth-root algorithm ==External links== * Special functions Elementary algebra

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Words begining with Cube_root:

Cube_root