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Cube Root


CubeRootReal
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CubeRootReImAbs
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Given a number z, the cube root of z, denoted RadicalBox[z, 3] or z^(1/3) (z to the 1/3 power), is a number a such that a^3=z. The cube root is therefore an nth root with n=3. Every real number has a unique real cube root, and every nonzero complex number has three distinct cube roots.

The schoolbook definition of the cube root of a negative number is (-x)^(1/3)=-(x^(1/3)). However, extension of the cube root into the complex plane gives a branch cut along the negative real axis for the principal value of the cube root as illustrated above. By convention, "the" (principal) cube root is therefore a complex number with positive imaginary part. As a result, the Wolfram Language and other symbolic algebra languages and programs that return results valid over the entire complex plane therefore return complex results for (-x)^(1/3). For example, in the Wolfram Language, ComplexExpand[(-1)^(1/3)] gives the result 1/2+isqrt(3)/2.

When considering a positive real number x, the Wolfram Language function CubeRoot[x], which is equivalent to Surd[x, 3], may be used to return the real cube root.

The cube root of a number a can be computed using Newton's method by iteratively applying

 x_n=1/3(a/(x_(n-1)^2)+2x_(n-1))

for some real starting value x_0.


See also

Cube Duplication, Cubed, Delian Constant, Geometric Problems of Antiquity, k-Matrix, nth Root, Radical, Root, Square Root, Surd

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Cite this as:

Weisstein, Eric W. "Cube Root." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubeRoot.html

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