Calculates the rational power using newton method

rational-pow.py

# Copyright Robert Eisele https://www.xarg.org/
from fractions import Fraction
import math

# Calculates (a/b)^(c/d) if solution is rational
def pow(a, b, c, d):

  xn = Fraction(int(math.pow(a, c / float(d))), int(math.pow(b, c / float(d))))
  abc = Fraction(a, b)**c

  for i in range(1, 6):
    xp = xn - (xn**d - abc) / (d * xn**(d - 1))
    if (xp == xn):
      return xn
    xn = xp
  return xn

print(pow(9, 1, 1, 2)) # == 3


# derivation:
# Root:   x = (a/b)^(c/d)
#   <=> x^d = (a/b)^c
#   <=>   0 = x^d - (a/b)^c
#  f(x) = x^d - (a/b)^c
# f'(x) = dx^(d-1)
# Newton method:
# xp = xn - f(xn) / f'(xn)