√³²•√⁶ √12

To simplify the expression \( \frac{\sqrt{32} \cdot \sqrt{6}}{\sqrt{12}} \), we can begin by simplifying each component under the square roots.

1. Simplify \( \sqrt{32} \): \[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]

2. Simplify \( \sqrt{6} \): \[ \sqrt{6} \text{ is already simplified.} \]

3. Simplify \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]

Now, substitute these simplifications back into the original expression:

\[ \frac{\sqrt{32} \cdot \sqrt{6}}{\sqrt{12}} = \frac{(4\sqrt{2}) \cdot \sqrt{6}}{2\sqrt{3}} \]

Next, calculate the numerator:

\[ 4\sqrt{2} \cdot \sqrt{6} = 4 \cdot \sqrt{2 \cdot 6} = 4\sqrt{12} \]

So now we rewrite the expression:

\[ \frac{4\sqrt{12}}{2\sqrt{3}} \]

Now simplify \( \frac{4}{2} \):

\[ = 2 \cdot \frac{\sqrt{12}}{\sqrt{3}} = 2 \cdot \sqrt{\frac{12}{3}} \]

Calculate \( \frac{12}{3} \):

\[ = 2 \cdot \sqrt{4} = 2 \cdot 2 = 4 \]

Thus, the final simplified result is:

\[ \boxed{4} \]