[tex]\frac{ 3 }{ 4 \sqrt{ 3 } +3 \sqrt{ 6 } } \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{\left(4\sqrt{3}+3\sqrt{6}\right)\left(4\sqrt{3}-3\sqrt{6}\right)} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{\left(4\sqrt{3}\right)^{2}-\left(3\sqrt{6}\right)^{2}} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{4^{2}\left(\sqrt{3}\right)^{2}-\left(3\sqrt{6}\right)^{2}} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{16\left(\sqrt{3}\right)^{2}-\left(3\sqrt{6}\right)^{2}} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{48-\left(3\sqrt{6}\right)^{2}} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{48-3^{2}\left(\sqrt{6}\right)^{2}} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{48-9\left(\sqrt{6}\right)^{2}} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{48-9\times 6} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{48-54} \\ = \frac{3\left(4\sqrt{3}-3\sqrt{6}\right)}{-6} \\ = -\frac{1}{2}\left(4\sqrt{3}-3\sqrt{6}\right) \\ = -\frac{1}{2}\times 4\sqrt{3}-\frac{1}{2}\left(-3\right)\sqrt{6} \\ \frac{-4}{2}\sqrt{3}-\frac{1}{2}\left(-3\right)\sqrt{6} \\ = -2\sqrt{3}-\frac{1}{2}\left(-3\right)\sqrt{6} \\ -2\sqrt{3}+\frac{-\left(-3\right)}{2}\sqrt{6} \\ = -2\sqrt{3}+\frac{3}{2}\sqrt{6} \\ = \frac{\sqrt{3}\left(3\sqrt{2}-4\right)}{2}[/tex]
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