Cube Root

Min Max

Min   Max       Re       Im

Given a number , the cube root of , denoted  or  ( to the 1/3 power), is a number  such that . The cube root is therefore an nth root with . 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 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 . For example, in the Wolfram Language, ComplexExpand[(-1)^(1/3)] gives the result .

When considering a positive real number , 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  can be computed using Newton's method by iteratively applying

for some real starting value .