<s>[INST] Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational. 
Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.

I've tried to attempt algebra on this problem. I noticed that there is some kind of nesting effect when trying to solve this. Please help me to understand how to attempt to denest this number.
Any help would be greatly appreciated.
 [/INST] Personally I like this method which is the same thing as the other answers.$\sqrt{4+2\sqrt{3}}=\sqrt{a}+\sqrt b$ squaring both sides $4+2\sqrt 3=a+2\sqrt{ab}+b$ for this to be true we must equate the parts with and without the radicals so $a+b=4$ and $2\sqrt{ab}=2\sqrt{3}$ so $ab=3$ now we have a system of equations that can be solved through substitution $a(4-a)=3\to a^2-4a+3=0$ which has roots  3, 1. So if a=1 then b=3 and vice versa if a=3. So our answer is $\sqrt{4+2\sqrt{3}}-\sqrt 3=1+\sqrt{3}-\sqrt3$
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