Simplify: $\frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}}$.
+248 \(\frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}}\)
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
\(\boxed{\frac{73\sqrt{2}}{152}}\)
