Min Max Register for Unlimited Interactive Examples >>

Min Max Re Register for Unlimited Interactive Examples >> Im

Given a number , the cube root of , denoted or ( to the 1/3 power), is a number such that . 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 . 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. By convention, "the" (principal) cube root is therefore a complex number with positive imaginary part. As a result, Mathematica and other symbolic algebra programs that return results valid over the entire complex plane therefore return complex results for . For example, in Mathematica, ComplexExpand[(-1)^(1/3)] gives the result . In versions of Mathematica prior to 6.0, this behavior could be changed by loading the package Miscellaneous`RealOnly`. The cube root of a number can be computed using Newton's method by iteratively applying

for some real starting value .

CITE THIS AS:

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