11. (40 points) Find all pairs of rational numbers $(a, b)$, for which $\sqrt{a}+\sqrt{b}=$ $=\sqrt{2+\sqrt{3}}$.
Answer: $(0.5 ; 1.5)$ and $(1.5 ; 0.5)$.

Solution. Squaring the equality, we get $a+b+2 \sqrt{a b}=2+\sqrt{3}$. Transferring $a+b$ to the right side and squaring again, we get $4 a b=(2-a-b)^{2}+2(2-a-b) \sqrt{3}+$ +3. In this expression, all terms except $2(2-a-b) \sqrt{3}$ are rational, and therefore this term must also be rational. This is possible only if $2-a-b=0$. Then we have $a+b=2, 2 \sqrt{a b}=\sqrt{3}$. Such a system is not difficult to solve: $b=2-a$, $a b=a(2-a)=\frac{3}{4}$. Solving the quadratic equation, we find both pairs of solutions.