<s>[INST] Determine if the number $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ is rational $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $
I have tried to raising it to the square, but I can't obtain the result.
$ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}}= k $
$ 2\sqrt{10+2\sqrt{5}} -2\sqrt{10+2\sqrt{5}} = k^2  $
$2(\sqrt{(10+2\sqrt{5})(10-2\sqrt{5}})=2\sqrt{80}=8\sqrt{5}=k^2$
Is this a good lead?
@EDIT one more thing, how to show that $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} + \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ equals to $\sqrt{10}+\sqrt{2}$
 [/INST] So, your steps aren't quite valid; if you have a statement of the form
$$
\sqrt{a} - \sqrt{b} = c
$$
Then, while it's true that 
$$
\left(\sqrt a - \sqrt b\right)^2 = c^2
$$
this unfortunately doesn't simplify to
$$
a - b = c^2
$$
rather, you get
$$
a - 2\sqrt a\sqrt b + b = c^2
$$
(check this by "foil"-ing).
What is true however, is that
$$
(\sqrt a - \sqrt b)(\sqrt a + \sqrt b) = a - b
$$
(again, check by "foil"-ing), and that $\sqrt a - \sqrt b$ is rational if and only if $\sqrt a + \sqrt b$ is (and so, as a consequence of this assumption, $a - b$ would be rational as well).
So
$$\left(\sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}}\right)\left(\sqrt{8+2\sqrt{10+2\sqrt{5}}} + \sqrt{8-2\sqrt{10+2\sqrt{5}}}\right)\\
= 8+2\sqrt{10+2\sqrt{5}} - (8-2\sqrt{10+2\sqrt{5}})\\
= 4\sqrt{10+2\sqrt{5}}$$
Now, you might try squaring things.
</s>