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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