To simplify the expression ( \frac{\sqrt{32} \cdot \sqrt{6}}{\sqrt{12}} ), we can begin by simplifying each component under the square roots.
Simplify ( \sqrt{32} ):
[
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
]
Simplify ( \sqrt{6} ):
[
\sqrt{6} \text{ is already simplified.}
]
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}
]
