Question
I saw a lot of problems that assume this: $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares. I wonder how can we demonstrate it because I saw a lot of people using it. Also, can I use it without a demonstration? Hope one of you can help me. Thank you!
My idea
$\sqrt{a}+\sqrt{b}=n$ where n is a natural number
$\sqrt{a}=n-\sqrt{b}$
$a=n^2+b-2n\sqrt{b}$
$\sqrt{b}=\frac{n^2+b-a}{2n}$
$\sqrt{b}=$ a rational number which means b is a perfect square.
I need to prove this, just so I can solve this problem Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?.
If you also find a way to solve this problem without using the fact that they must be perfect squares. I will be greatful.