[tex] \frac{2}{ \sqrt{8 + 2 \sqrt{2} } } = \frac{2}{ \sqrt{8 + 2 \sqrt{2} } } \times \frac{ \sqrt{8 - 2 \sqrt{2} } }{ \sqrt{8 - 2 \sqrt{2} } } \\ = \frac{2 \sqrt{8 - 2 \sqrt{2} } }{ \sqrt{64 - 8} } \\ = \frac{2 \sqrt{8 - 2 \sqrt{2} } }{ \sqrt{56} } \\ = \frac{2 \sqrt{8 - 2 \sqrt{2} } }{ \sqrt{4 \times 14} } \\ = \frac{2 \sqrt{8 - 2 \sqrt{2} } }{2 \sqrt{14} } \\ = \frac{ \sqrt{2(4 - \sqrt{2}) } }{ \sqrt{2 \times 7} } \\ \: \: \: \: \: \: \: = \frac{ \sqrt{4 - \sqrt{2} } }{ \sqrt{7} } \times \frac{ \sqrt{7} }{ \sqrt{7} } \\ = \frac{ \sqrt{7(4 - \sqrt{2}) } }{7} \\ = \frac{ \sqrt{28 - 7\sqrt{2} } }{7} [/tex]
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