$\sqrt{8+2\sqrt{7}}=\sqrt{7+1+2\sqrt{7}}=\sqrt{7+2\sqrt{7}+1}=\sqrt{(\sqrt{7}{)}^2+2\ast <!--\; \ast \; -->1\ast <!--\; \ast \; -->\sqrt{7}+1^2}=\sqrt{(\sqrt{7}+1{)}^2}=|\sqrt{7}+1|=\sqrt{7}+1$

$\sqrt{15+6\sqrt{6}}=\sqrt{6+9+6\sqrt{6}}=\sqrt{6+6\sqrt{6}+9}=\sqrt{(\sqrt{6}{)}^2+2\ast <!--\; \ast \; -->3\ast <!--\; \ast \; -->\sqrt{6}+3^2}=\sqrt{(\sqrt{6}+3{)}^2}=|\sqrt{6}+3|=\sqrt{6}+3$

$\sqrt{7+2\sqrt{10}}=\sqrt{2+5+2\sqrt{10}}=\sqrt{2+2\sqrt{10}+5}=\sqrt{(\sqrt{2}{)}^2+2\sqrt{2\ast <!--\; \ast \; -->5}+(\sqrt{5}{)}^2}=\sqrt{(\sqrt{2}+\sqrt{5}{)}^2}=|\sqrt{2}+\sqrt{5}|=\sqrt{2}+\sqrt{5}$