[tex]~~~~\dfrac{5+2\sqrt 3}{7+4\sqrt 3} = a- 6\sqrt 3\\\\\\\implies a= \dfrac{5+2\sqrt 3}{7+4\sqrt 3}+6\sqrt3 \\\\\\\implies a=\dfrac{\left(5+2\sqrt 3 \right) \left( 7-4\sqrt 3\right)}{\left(7+4\sqrt 3 \right) \left(7-4\sqrt 3 \right)}+6\sqrt 3\\\\\\\implies a = \dfrac{35-20\sqrt 3+14\sqrt 3-24}{7^2 - \left(4\sqrt 3 \right)^2}+6\sqrt 3\\\\\\\implies a=\dfrac{11-6\sqrt 3}{49-48}+6\sqrt 3\\\\\\\implies a=11-6\sqrt 3+6\sqrt 3\\\\\\\implies a = 11[/tex]
