Surd inside Surd equality

Question

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?

Answer 1

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 $\sqrt a-\sqrt b$.)

Answer 2

If we square $$\sqrt{6 - 2 \sqrt{5} }$$ We get $$6- 2 \sqrt{5}=(\sqrt 5 -1)^2$$

Thus $$\sqrt{6 - 2 \sqrt{5} } = \sqrt 5 -1$$