Algebra question

Simplify

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\(\large \frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{2}}}} \) = \(\frac{1}{\sqrt{2}+\frac{1}{\frac{\sqrt{8\cdot 2}+\sqrt{200\cdot 2}+1}{\sqrt{2}}}}\) 

= \(\frac{1}{\sqrt{2}+\frac{\sqrt{2}}{\sqrt{8\cdot 2}+\sqrt{200\cdot 2}+1}}\) = \(\frac{\sqrt{8\cdot 2}+\sqrt{200\cdot 2}+1}{(\sqrt{8\cdot 2}+\sqrt{200\cdot 2}+1)\cdot \sqrt{2}+\sqrt{2} }\) = \(\frac{4+20+1}{\sqrt{2}(16+20+1+1)}\) 

= \(\frac{25}{26\sqrt{2}}\color{blue}=\frac{25\sqrt{2}}{52}\)

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