rationalizing denominators

Annoying - it boils down to $\boxed{\frac{73\sqrt{2}}{152}}$.

Correct thedudemanguyperson...!

$\frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}}$

                                        $​​​​\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}=\frac{3\sqrt{2}}{73}$

$=\frac{1}{\sqrt{2}+\frac{3\sqrt{2}}{73}}$

                                        $\sqrt{2}+\frac{3\sqrt{2}}{73}:{\frac{76\sqrt{2}}{73}}$                  

$=\frac{1}{\frac{76\sqrt{2}}{73}}$

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

$=\frac{73}{76\sqrt{2}}$

                                      $Rationalize \frac{73}{76\sqrt{2}}:\frac{73\sqrt{2}}{152}$

$=\frac{73\sqrt{2}}{152}$