Correct thedudemanguyperson...!

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

\(\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}=\frac{3\sqrt{2}}{73} \)

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

\(\sqrt{2}+\frac{3\sqrt{2}}{73}:{\frac{76\sqrt{2}}{73}}\)

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

\(\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{1}{\frac{b}{c}}=\frac{c}{b}\)

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

\(Rationalize \frac{73}{76\sqrt{2}}:\frac{73\sqrt{2}}{152}\)

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