Question

Find the value of x given that

Answer 1

For the first part, the results get quite crazy. I hardly believe that you're supposed to do this by hand, so I'm afraid that there's some typo in the question.

As for the second part, we have

$\sqrt{x}+\dfrac{1}{\sqrt{x}}=\dfrac{x+1}{\sqrt{x}}$

Plug your value for x and you have

$\dfrac{x+1}{\sqrt{x}} \mapsto \dfrac{4+2\sqrt{2}}{\sqrt{3+2\sqrt{2}}}=\dfrac{4+2\sqrt{2}}{1+\sqrt{2}}$

We can rewrite the quantity as

$\dfrac{4+2\sqrt{2}}{1+\sqrt{2}}\cdot\dfrac{1-\sqrt{2}}{1-\sqrt{2}}=\dfrac{4-4\sqrt{2}+2\sqrt{2}-4}{-1} = 2\sqrt{2}$