My argument uses three assumptions that are likely true:
1. The cube root of an integer that is not a perfect cube is irrational.
2. The product of a nonzero rational number and an irrational number is irrational.
3. The sum of two irrational numbers is also irrational.
Suppose x + cubrt(2)*y + cubrt(4)*z = 0 has a nontrivial solution, in that at least one of x, y, z is nonzero. Then -x = cubrt(2)*y + cubrt(4)*z, which is irrational by the above three assumptions. But x is rational, so -x must also be rational -- a contradiction. Therefore the original equation has no nontrivial solution, hence x=y=z=0 is the only solution.
Edited on October 15, 2009, 4:00 pm