smc

We have $$\sqrt{1}+\sqrt{1+3}+\sqrt{1+3+5}+\sqrt{1+3+5+7}=\sqrt{1}+\sqrt{4}+\sqrt{9}+\sqrt{16}=1+2+3+4=\boxed{10}.$$ Note: This comes from the fact that the sum of the first $n$ odds is $n^2$.