2150.81 – A Horrendous Quantity

Here is the horrendous thing:

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

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

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

$m = \sqrt{3 - 2 \sqrt{2}} = \sqrt{2 - 2 \sqrt{2} + 1} = \sqrt{(\sqrt{2})^2 - 2 \sqrt{2} + 1} = \sqrt{(\sqrt{2} - 1)^2} = \sqrt{2} - 1.$

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