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

$\sqrt{15 + 6\sqrt{6}} = \sqrt{6 + 9 + 6\sqrt{6}} = \sqrt{6 + 6\sqrt{6} + 9} = \sqrt{(\sqrt{6})^2 + 2 * 3 * \sqrt{6} + 3^2} = \sqrt{(\sqrt{6} + 3)^2} = |\sqrt{6} + 3| = \sqrt{6} + 3$

$\sqrt{7 + 2\sqrt{10}} = \sqrt{2 + 5 + 2\sqrt{10}} = \sqrt{2 + 2\sqrt{10} + 5} = \sqrt{(\sqrt{2})^2 + 2\sqrt{2 * 5} + (\sqrt{5})^2} = \sqrt{(\sqrt{2} + \sqrt{5})^2} = |\sqrt{2} + \sqrt{5}| = \sqrt{2} + \sqrt{5}$