jmc

Notice that multiplying all three of the original equations together tells us that $(a^2{b}^2{c}^2)/(abc)=6$, which implies $abc=6$. Rewriting the first and third equations as $b=a^2$ and $c=\sqrt{3a}$ and plugging these into $abc=6$ yields $a\cdot a^2\cdot \sqrt{3a}=6$. By squaring both sides of the equation, we obtain $3a^7=36\Rightarrow a=\boxed{12^{1/7}}$.