Which is greater?

Squaring the three equations we get,

$({\sqrt{7}+\sqrt{10})}^2=17+2\sqrt{70}$ .........................(1)

$({\sqrt{3}+\sqrt{19})}^2=22+2\sqrt{57}$ .........................(2)

$({\sqrt{8}+\sqrt{9})}^2=17+2\sqrt{72}$ ............................(3)

Reduce the expressions (1) ,(2) and (3) by 17 and we have,

2 $\sqrt{70}$ , 5+ 2 $\sqrt{57}$ and 2 $\sqrt{72}$ respectively .

Square these expressions.This yields

${(2\sqrt{70})}^2 =280$

${(5+2\sqrt{57})}^2 =253+20\sqrt{57}$

${(2\sqrt{72})}^2 =288$

Subtract 253 from each and we get,

27 , $20\sqrt{57}$ and 35 respectively

Since $\sqrt{57}$ is greater than 2 , it follows that $20\sqrt{57}$ > 35 > 27

Hence $(\sqrt{3}+\sqrt{19})$ is the greatest expression.

$\begin{cases} (\sqrt{7}+\sqrt{10})^2 & = 17+2\sqrt{70} \\ (\sqrt{3}+\sqrt{19})^2 & = 22+2\sqrt{57} \\ (\sqrt{8}+\sqrt{9})^2 & = 17+2\sqrt{72} \end{cases} \quad \Rightarrow \sqrt{8}+\sqrt{9} > \sqrt{7}+\sqrt{10}$

We note that:

$(\sqrt{8}+\sqrt{9})^2 = 17+2\sqrt{72} = 17+12\sqrt{2} < 17+12(1.5) = 35$

$(\sqrt{3}+\sqrt{19})^2 = 22+2\sqrt{57} > 22+2\sqrt{49} = 36$

Therefore, $\boxed{\sqrt{3}+\sqrt{19}} > \sqrt{8}+\sqrt{9} > \sqrt{7}+\sqrt{10}$