Can the sum of two irrational roots of two coprime natural numbers greater than 1 be an irrational root of a rational number? After thinking more about my last question and reading the answers I reformulated it. I'm positive I wanted to ask the following:  
Let $x,y\in \mathbb{N};\: x,y>1;\: gcd(x,y)=1; \: \sqrt{x},\sqrt{y}\notin\mathbb{N}$ and let $z\in\mathbb{Q}; \: \sqrt{z}\notin\mathbb{Q}$.
Is it possible to find $x, y, z$ obeying the rules above such that the following equality is true?  
$\sqrt{x}+\sqrt{y}=\sqrt{z}$ 
If so, find a sufficient and necessary condition in the choosing of $x$ and $y$ for $z$ to exist. 
Basically I want to know if we can find $r$ rational which makes the equality true in expressions like $\sqrt{2}+\sqrt{3}=\sqrt{r} \: and \: \sqrt{5}+\sqrt{11}=\sqrt{r}$.

Hint:
If $\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{z}$  then squaring both sides gives
$$x+y+2\sqrt{xy}=z$$
Note that $x+y\in \Bbb{N}$ and $z\in \Bbb{Q}$. 
So if $\gcd(x,y)=1$ then $\sqrt{xy}\in\Bbb{R} \setminus\Bbb{Q}$
So what do we infer from this?
