simplify: 1/2+root 5+ 1/root 5+root 6+ 1/root 6+root 7+ 1/root 7+root 8

\frac{1}{2 + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} = \frac{1}{\sqrt{5} + 2} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{7} + \sqrt{6}} + \frac{1}{\sqrt{8} + \sqrt{7}} Now, \frac{1}{\sqrt{5} + 2} = \frac{1}{\sqrt{5} + 2} \times \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{5 - 4} = \sqrt{5} - 2 \frac{1}{\sqrt{6} + \sqrt{5}} = \frac{1}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} = \frac{\sqrt{6} - \sqrt{5}}{6 - 5} = \sqrt{6} - \sqrt{5} \frac{1}{\sqrt{7} + \sqrt{6}} = \frac{1}{\sqrt{7} + \sqrt{6}} \times \frac{\sqrt{7} - \sqrt{6}}{\sqrt{7} - \sqrt{6}} = \frac{\sqrt{7} - \sqrt{6}}{7 - 6} = \sqrt{7} - \sqrt{6} \frac{1}{\sqrt{8} + \sqrt{7}} = \frac{1}{\sqrt{8} + \sqrt{7}} \times \frac{\sqrt{8} - \sqrt{7}}{\sqrt{8} - \sqrt{7}} = \frac{\sqrt{8} - \sqrt{7}}{8 - 7} = \sqrt{8} - \sqrt{7} Now, \frac{1}{\sqrt{5} + 2} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{7} + \sqrt{6}} + \frac{1}{\sqrt{8} + \sqrt{7}} = (\sqrt{5} - 2) + (\sqrt{6} - \sqrt{5}) + (\sqrt{7} - \sqrt{6}) + (\sqrt{8} - \sqrt{7}) = \sqrt{5} - 2 + \sqrt{6} - \sqrt{5} + \sqrt{7} - \sqrt{6} + \sqrt{8} - \sqrt{7} = \sqrt{8} - 2 = 2 \sqrt{2} - 2 = 2 (\sqrt{2} - 1)

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