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.

Let $x=\sqrt{4+2\sqrt{3}}-\sqrt{3}$. Then:
$$(x+\sqrt3)^2=4+2\sqrt{3}\implies x^2-1=2(1-x)\sqrt3\implies(x-1)(x+1+2\sqrt3)=0$$
So, certainly $x=1$ or $x=-1-2\sqrt3$. But a moment's thought (e.g. considering that $x>0$) convinces us that the first of these must be the case - i.e., $x=1$ (a known rational number).
