\[\left( 3\sqrt{12} + \frac{1}{2}\sqrt{48} \right) \cdot \sqrt{3} =\]
\[= \left( 3 \cdot 2\sqrt{3} + \frac{1}{2} \cdot 4\sqrt{3} \right) \cdot \sqrt{3} =\]
\[= \left( 6\sqrt{3} + 2\sqrt{3} \right) \cdot \sqrt{3} =\]
\[= 8\sqrt{3} \cdot \sqrt{3} = 8 \cdot 3 = 24.\]