\(\frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}}\)

First, simplify the radicles in the expression

\(\sqrt{8}=\sqrt{2 \cdot 2\cdot 2} =2\sqrt{2}\)

\(\sqrt{200}=\sqrt{10\cdot20\cdot2}=10\sqrt{2}\)

\(\sqrt{18}=\sqrt{3\cdot3\cdot2}=3\sqrt{2}\)

\(\frac{1}{\sqrt{2}+\frac{1}{2\sqrt{2}+10\sqrt{2}+\frac{1}{3\sqrt{2}}}}\)

The two radicles can be added together because they have the same base

\(\frac{1}{\sqrt{2}+\frac{1}{12\sqrt{2}+\frac{1}{3\sqrt{2}}}}\)

Now start simplifying from the innermost fraction

Find a common denominator to add the two on the bottom

\(\frac{1}{\sqrt{2}+\frac{1}{\frac{73}{3\sqrt{2}}}}\)

\(\frac{1}{\sqrt{2}+\frac{3\sqrt{2}}{73}}\)

Find a common denominator again

\(\frac{1}{\frac{73\sqrt{2}}{73}+\frac{3\sqrt{2}}{73}}\)

Add the fractions

\(\frac{1}{\frac{76\sqrt{2}}{73}}\)

\(\frac{73}{76\sqrt{2}}\)

Multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\) to get the radicle out of the denominator

\(\boxed{\frac{73\sqrt{2}}{152}}\)