Ta có:
$S=a^2+\dfrac{36\sqrt6}{\sqrt a}+\dfrac{36\sqrt6}{\sqrt a}+\dfrac{36\sqrt6}{\sqrt a}+\dfrac{36\sqrt6}{\sqrt a}-\dfrac{144\sqrt6-18}{\sqrt a}$
$\ge 5\sqrt[5]{a^2.\dfrac{36\sqrt6}{\sqrt a}.\dfrac{36\sqrt6}{\sqrt a}.\dfrac{36\sqrt6}{\sqrt a}.\dfrac{36\sqrt6}{\sqrt a}}-\dfrac{144\sqrt6-18}{\sqrt 6}$
$=36+3\sqrt6$
Vậy $\min S=36+3\sqrt6 \Leftrightarrow a=6$