Solve this equation

x^3/z^3 + y^3/z^3 = 6

x,y, and z are positive coprime integers. (Coprime meaning that they have no common divisor aside from 1)

This challenge appeared in a 1994 edition of the London Sunday Telegraph with a cash reward of 450 pounds, but with the added stipulation that x,y,and z must be greater than 100. Since the reward here is merely a sincere attaboy!, I am dropping the >100 requirement.