\(\left(12-6\sqrt{3}\right)\sqrt{\dfrac{3}{14-8\sqrt{3}}}-3\sqrt{2\left(1-\sqrt{-2\sqrt{3}+4}\right)+2\sqrt{4+2\sqrt{3}}}\)
\(=\left(12-6\sqrt{3}\right).\dfrac{\sqrt{3}}{\sqrt{14-8\sqrt{3}}}-3\sqrt{2\left(1-\sqrt{\left(\sqrt{3}-1\right)^2}\right)+2\sqrt{\left(\sqrt{3}+1\right)^2}}\)
\(=\left(12-6\sqrt{3}\right).\dfrac{\sqrt{3}}{\sqrt{14+2\sqrt{48}}}-3\sqrt{2\left(1-\left|\sqrt{3}-1\right|\right)+2\left|\sqrt{3}+1\right|}\)
\(=\left(12-6\sqrt{3}\right).\dfrac{\sqrt{3}}{\sqrt{\left(\sqrt{8}+\sqrt{6}\right)^2}}-3\sqrt{2\left(1-\sqrt{3}+1\right)+2\sqrt{3}+2}\)
\(=\dfrac{12\sqrt{3}-18}{\left|\sqrt{8}+\sqrt{6}\right|}-3\sqrt{2\left(2-\sqrt{3}\right)+2\sqrt{3}+2}\)
\(=\dfrac{12\sqrt{3}-18}{\sqrt{8}+\sqrt{6}}-3\sqrt{4-2\sqrt{3}+2\sqrt{3}+2}\)
\(=\dfrac{12\sqrt{3}-18}{\sqrt{8}+\sqrt{6}}-3\sqrt{6}\\ =\dfrac{12\sqrt{3}-18-3\sqrt{6}\left(\sqrt{8}+\sqrt{6}\right)}{\sqrt{8}+\sqrt{6}}\\ =\dfrac{12\sqrt{3}-18-18-12\sqrt{3}}{\sqrt{8}+\sqrt{6}}\\ =\dfrac{-36}{\sqrt{8}+\sqrt{6}}\)
\(=\left(12-6\sqrt{3}\right)\cdot\sqrt{\dfrac{21+12\sqrt{3}}{2}}-3\cdot\sqrt{2\cdot\left(1-\sqrt{1-\sqrt{3}+1}\right)+2\sqrt{3}+2}\)
\(=\left(12-6\sqrt{3}\right)\cdot\dfrac{2\sqrt{3}+3}{\sqrt{2}}-3\cdot\sqrt{2-\sqrt{2}\left(\sqrt{4-2\sqrt{3}}\right)+2\left(\sqrt{3}+1\right)}\)
\(=3\sqrt{6}-3\cdot\sqrt{2-\sqrt{2}\left(\sqrt{3}-1\right)+2\sqrt{3}+2}\)
\(=3\sqrt{6}-3\cdot\sqrt{4+2\sqrt{3}-\sqrt{6}+\sqrt{2}}\)
