<s>[INST] Surd inside Surd equality I was trying a problem where I got the following surd as my answer:
$$ {\sqrt{6 - 2 \sqrt{5} }\over 4} \approx 0.309016.... $$
The answer listed was:
$$ {\sqrt{5} - 1 \over 4} \approx 0.309016....  $$
Is there a simple way that you could get to the second expression from the first? 
 [/INST] Suppose you assume that you can write $\sqrt{6-2\sqrt5} = \sqrt a - \sqrt b$ for some rational numbers $a$ and $b$. Squaring both sides will give you $6-2\sqrt5 = a+b-2\sqrt{ab}$, and so
\begin{align*}
6 &= a+b \\
\sqrt{5} &= \sqrt{ab}
\end{align*}
So we want $ab=5$ and $a+b=6$. An obvious solution is, indeed, $a=5$ and $b=1$. (Taking $a=1$ and $b=5$ gives a negative result for $\sqrt a-\sqrt b$.)
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