Simplify $\frac{1}{\sqrt{4+2\sqrt{3}} - \sqrt{4-2\sqrt{3}}}$ Simplify $$\frac{1}{\sqrt{4+2\sqrt{3}} - \sqrt{4-2\sqrt{3}}}$$
I know there is another easier method except the one I answered. I cannot find it. Can you please help? Thanks in advance.

Let $\zeta_{\pm}=\sqrt{4\pm 2\sqrt{3}}$. We have:
$$ \frac{1}{\zeta_{+}-\zeta_{-}}=\frac{\zeta_{+}+\zeta_{-}}{\zeta_{+}^2-\zeta_{-}^2}=\sqrt{\frac{\zeta_{+}^2+\zeta_{-}^2+2\zeta_{+}\zeta_{-}}{(\zeta_{+}^2-\zeta_{-}^2)^2}} $$
but $\zeta_{+}\zeta_{-}=2$, $\zeta_{+}^2-\zeta_{-}^2=4\sqrt{3}$ and $\zeta_{+}^2+\zeta_{-}^2=8$, hence:
$$ \frac{1}{\zeta_{+}-\zeta_{-}}=\sqrt{\frac{8+4}{48}}=\color{red}{\frac{1}{2}}.$$
