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

$\sqrt{5-2\sqrt{6}}=\sqrt{3+2-2\sqrt{6}}=\sqrt{(\sqrt{3}{)}^2-2\sqrt{3\ast 2}+(\sqrt{2}{)}^2}=\sqrt{(\sqrt{3}-\sqrt{2}{)}^2}=|\sqrt{3}-\sqrt{2}|=\sqrt{3}-\sqrt{2}$

$\sqrt{12-6\sqrt{3}}=\sqrt{9+3-6\sqrt{3}}=\sqrt{3^2-2\ast 3\ast \sqrt{3}+(\sqrt{3}{)}^2}=\sqrt{(3-\sqrt{3}{)}^2}=|3-\sqrt{3}|=3-\sqrt{3}$

$\sqrt{38-12\sqrt{2}}=\sqrt{36+2-12\sqrt{2}}=\sqrt{6^2-2\ast 6\ast \sqrt{2}+(\sqrt{2}{)}^2}=\sqrt{(6-\sqrt{2}{)}^2}=|6-\sqrt{2}|=6-\sqrt{2}$