Math2.org Math Tables: Recursive Formulas for y^1/n

Recursive Formulas for y^1/n

Explicit form:

y^1/n = x

Recursive form:

x_k+1 = (x_k + y / (x_k)^n-1) / 2

where y ³ 0 and n > 0
or y Î Â and n is odd, positive, and integer.
(for negative n, evaluate the above formula with n positive, then invert your answer).

Example: 2^1/3 = 1.259921049894...

iteration	value
x0	1.0000000000
x1	1.5000000000
x2	1.1944444444
x3	1.2981416381
x4	1.2424821566
x5	1.2690093603
x6	1.2554742937
x7	1.2621680807
x8	1.2588035314
x9	1.2604812976
x10	1.2596412994
x20	1.2599207765
x50	1.259921049894
x	1.259921049894...

Source: Jeff Yates, et al.