Find conditions on positive integers $a, b, c$ so that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is irrational.
My solution: if $ab$ is not the square of an integer, then the expression is irrational. I find it interesting that $c$ does not come into this at all.
My solution is modeled (i.e., copied with modifications) from dexter04's solution to Prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational .
Suppose $\sqrt{a}+\sqrt{b}+\sqrt{c} = r$ where $r$ is rational. Then, $(\sqrt{a}+\sqrt{b})^2 = (r-\sqrt{c})^2 \implies a+b+2\sqrt{ab} = c+r^2-2r\sqrt{c}$.
So, $a+b-c-r^2+2\sqrt{ab} =-2r\sqrt{c}$. Let $a+b-c-r^2 = k$, which will be a rational number. So, $(k+2\sqrt{ab})^2 = k^2+ 4ab+4k\sqrt{ab} = 4cr^2$ or $4k\sqrt{ab} = 4cr^2-k^2- 4ab$.
If $ab$ is not a square of an integer, then the LHS is irrational while the RHS is rational. Hence, we have a contradiction.