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.
You could 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 been proved that if Öa is
rational then a must be a square number.
See the problem The
Root Cause .
Try to apply this method and then to extend it to three variables for the
last part.