In this problem you are given that a, b and c are natural numbers.
You have to show that if √a+√b is rational then it is a natural number.
Perhaps you can crack this Tough Nut if you use the fact that if √a+√b is rational then so is its square which means that √ab is also rational. Knowing this the next step is to use
to show that √a is rational and to do likewise for b.
This is all you need because it has already been proved that if √a 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
√ab + √bc +√ca and √abc.