If $a,b,c$ are rational numbers and $a\sqrt2+b\sqrt3+c\sqrt5=0$, show that $a=b=c=0$ 
If $a,b,c$ are rational numbers and $a\sqrt2+b\sqrt3+c\sqrt5=0$, show that $a=b=c=0$

I can solve for two terms only but for additional $c\sqrt5$ term, I got stuck. I also want to know how to solve this for general $n$ terms i.e. $a\sqrt2+b\sqrt3+c\sqrt5+\ldots+n \text{ terms}=0$  will imply $a=b=c=\ldots=0$

Multiply your equation by $\sqrt{2}$ to get,
$2a + \sqrt{6}b + \sqrt{10}c = 0 \implies a = 0,  \sqrt{6}b + \sqrt{10}c = 0$ since the rational and irrational parts both need to be zero.
Now multiply $\sqrt{6}b + \sqrt{10}c = 0$ by $\sqrt{6}$ and apply the same logic. 
