Unchanged Conjugate Radical? (Rationalizing Demoninators)

Question

I'm on one of the more difficult practice problems on Excercise 1-6 in "AoPS:Vol. 1".

Problem £4 : Ex. 1-6. The hint details that we should multiply $$\left(\frac{1}{\sqrt 1 \sqrt 2}\right) $$ by it's conjugate which the hint identifies as itself.

I have tried rationalizing the above fraction by multiplying it's conjugate by itself, however, how can we FOIL the two terms when there appears to only be one?

Overall Questions: 1. How the Conjugate Radical be the same as the original expression? If so, why? 2. How can we rationalize a fraction like this which appears to have only one term, when rationalization by conjugate radicals always requires two?

Thanks.

Answer 1

$1/\sqrt{1+\sqrt{2}} = \sqrt{1+\sqrt{2}}/(1+\sqrt{2})$.

Now multiply top and bottom by $1-\sqrt(2)$.

$\sqrt{1+\sqrt{2}}(1-\sqrt(2))/((1-\sqrt(2))((1+\sqrt{2}))$

$=\sqrt{1+\sqrt{2}}(1-\sqrt(2))/(-1)$

$=\sqrt{1+\sqrt{2}}(\sqrt(2)-1)$