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.

Such square roots can  be computed by a very Simple Denesting Rule:
Here $\, 4+2\sqrt 3\ $ has norm $= 4.\:$ $\rm\ \color{blue}{Subtracting\ out}\,\ \sqrt{norm}\ = 2\,\ $ yields $\,\  2+2\sqrt 3\:$
which has $\, {\rm\ \sqrt{trace}}\, =\, \sqrt{4}\, =\, 2.\,\ \  \rm \color{brown}{Dividing\ it\ out}\,\ $ of the above  yields $\ \ 1+\sqrt 3$ 
Remark $\ $ Many more worked examples are in prior posts on this denesting rule.
