12√8* √3/2

To simplify the expression \( 12\sqrt{8} \cdot \frac{\sqrt{3}}{2} \), we can follow these steps: 1. **Simplify \( \sqrt{8} \)**: \[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \] 2. **Substitute back into the expression**: \[ 12\sqrt{8} = 12 \cdot 2\sqrt{2} = 24\sqrt{2} \] 3. **Now substitute this back into the expression**: \[ 24\sqrt{2} \cdot \frac{\sqrt{3}}{2} \] 4. **Multiply by \(\frac{1}{2}\)**: \[ = 24\sqrt{2} \cdot \frac{\sqrt{3}}{2} = \frac{24\sqrt{2} \cdot \sqrt{3}}{2} \] 5. **Now simplify the fraction**: \[ = 12\sqrt{6} \] So, the final simplified result of \( 12\sqrt{8} \cdot \frac{\sqrt{3}}{2} \) is: \[ \boxed{12\sqrt{6}} \]