You could use the fact that if $\sqrt{a}+\sqrt{b}$ is rational then so is its square which means that $\sqrt{a}b$ is also rational. Knowing this the next step is to use

${\displaystyle \sqrt{a}(\sqrt{a}+\sqrt{b})=a+\sqrt{\mathrm{ab}}}$

to show that $\sqrt{a}$ is rational and to do likewise for $b$.

This is all you need because it has been proved that if $\sqrt{a}$ 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.