2150.81 – A Horrendous Quantity

Here is the horrendous thing:

$$N = \dfrac{\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}}{\sqrt{\sqrt{5} + 1}} + \sqrt{3-2 \sqrt{2}} = k - m.$$

We will deal with the two terms separately. The first term, which we call $k$, we will square:

$$\begin{aligned}k^2 &= \left( \dfrac{\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}}{\sqrt{\sqrt{5} + 1}} \right)^2 \\[1em]&= \dfrac{(\sqrt{5} + 2) + 2 \sqrt{(\sqrt{5} + 2)(\sqrt{5} - 2)} + (\sqrt{5} - 2)}{\sqrt{5} + 1)} \\[1em]&= \dfrac{2 \sqrt{5} + 2 \sqrt{5-4}}{\sqrt{5} + 1} \\[1em]&= \dfrac{2 \sqrt{5} + 2}{\sqrt{5} + 1} = 2\end{aligned}$$

So $k = \sqrt{2}$. The second term, $m$ does not require squaring:

$$\begin{aligned}m &= \sqrt{3 - 2 \sqrt{2}} \\&= \sqrt{2 - 2 \sqrt{2} + 1} \\&= \sqrt{(\sqrt{2})^2 - 2 \sqrt{2} + 1} \\&= \sqrt{(\sqrt{2} - 1)^2} \\&= \sqrt{2} - 1\end{aligned}$$

Thus $N = k - m = 1$. Surprise!