Question

Simplify . Give the answer in steps

Answer 1

A familiar question to me this is! And also my favourite! 

$\frac{1}{3} - \sqrt{8} - \frac{1}{\sqrt{8} - \sqrt{7}} + \frac{1}{\sqrt{7} - \sqrt{6}} - \frac{1}{\sqrt{6} - \sqrt{5}} + \frac{1}{\sqrt{5} - 2} \\ \\ \\ \frac{1}{\sqrt{8} - \sqrt{7}} = \frac{1(\sqrt{8} + \sqrt{7})}{(\sqrt{8} - \sqrt{7})(\sqrt{8} + \sqrt{7})} = \frac{\sqrt{8} + \sqrt{7}}{(\sqrt{8})^2 - (\sqrt{7})^2} = \frac{\sqrt{8} + \sqrt{7}}{8 - 7} = \frac{\sqrt{8} + \sqrt{7}}{1} = \sqrt{8} + \sqrt{7}$ 

$\frac{1}{\sqrt{7} - \sqrt{6}} = \frac{1(\sqrt{7} + \sqrt{6})}{(\sqrt{7} - \sqrt{6})(\sqrt{7} + \sqrt{6})} = \frac{\sqrt{7} + \sqrt{6}}{(\sqrt{7})^2 - (\sqrt{6})^2} = \frac{\sqrt{7} + \sqrt{6}}{7 - 6} = \frac{\sqrt{7} + \sqrt{6}}{1} = \sqrt{7} + \sqrt{6}$ 

$\frac{1}{\sqrt{6} - \sqrt{5}} = \frac{1(\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})} = \frac{\sqrt{6} + \sqrt{5}}{(\sqrt{6})^2 - (\sqrt{5})^2} = \frac{\sqrt{6} + \sqrt{5}}{6 - 5} = \frac{\sqrt{6} + \sqrt{5}}{1} = \sqrt{6} + \sqrt{5}$ 

$\frac{1}{\sqrt{5} - 2} = \frac{1(\sqrt{5} + 2)}{(\sqrt{5} - 2)(\sqrt{5} + 2)} = \frac{\sqrt{5} + 2}{(\sqrt{5})^2 - 2^2} = \frac{\sqrt{5} + 2}{5 - 4} = \frac{\sqrt{5} + 2}{1} = \sqrt{5} + 2$ 

$\frac{1}{3} - \sqrt{8} - \frac{1}{\sqrt{8} - \sqrt{7}} + \frac{1}{\sqrt{7} - \sqrt{6}} - \frac{1}{\sqrt{6} - \sqrt{5}} + \frac{1}{\sqrt{5} - 2} \\ \\ = \frac{1}{3} - \sqrt{8} - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + 2) \\ \\ = \frac{1}{3} - \sqrt{8} - \sqrt{8} - \sqrt{7} + \sqrt{7} + \sqrt{6} - \sqrt{6} - \sqrt{5} + \sqrt{5} + 2 \\ \\ = \frac{1}{3} - 2\sqrt{8} + 2 = \frac{1}{3} - 4\sqrt{2} + 2 = \frac{7}{3} - 4\sqrt{2} = \frac{7 - 12\sqrt{2}}{3}$ 

If the question was like $\frac{1}{3} + \sqrt{8} - \frac{1}{\sqrt{8} - \sqrt{7}} + \frac{1}{\sqrt{7} - \sqrt{6}} - \frac{1}{\sqrt{6} - \sqrt{5}} + \frac{1}{\sqrt{5} - 2}$, 

$\frac{1}{3} + \sqrt{8} - \frac{1}{\sqrt{8} - \sqrt{7}} + \frac{1}{\sqrt{7} - \sqrt{6}} - \frac{1}{\sqrt{6} - \sqrt{5}} + \frac{1}{\sqrt{5} - 2} \\ \\ = \frac{1}{3} + \sqrt{8} - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} + 2) \\ \\ = \frac{1}{3} + \sqrt{8} - \sqrt{8} - \sqrt{7} + \sqrt{7} + \sqrt{6} - \sqrt{6} - \sqrt{5} + \sqrt{5} + 2 \\ \\ = \frac{1}{3} + 2 = \frac{7}{3}$ 

Hope this may be helpful. 

Please mark my answer as the brainliest if this helps you. 

Thank you. Have a nice day.