In this problem you are given that a, b and c are natural numbers.
You have to show that if Öa+Öb is rational then it is a natural number.
Perhaps you can crack this Tough Nut if you use the fact that if Öa+Öb is rational then so is its square which means that Öab is also rational. Knowing this the next step is to use
to show that Öa is rational and to do likewise for b.
This is all you need because it has already been proved that if Öa is rational it must be an integer.
See the problem The
Root Cause in the June 2000 15+ Challenges.
Try to apply this method and then to extend it to three variables for the last part. You will need to consider
Öab + Öbc +Öca and Öabc.