To find the product of \(\sqrt{12}\) and \(56\), we first compute \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Next, we multiply this result by \(56\):
\[ 56 \cdot \sqrt{12} = 56 \cdot (2\sqrt{3}) = 112\sqrt{3} \]
Now, we need to find a numerical value for \(112\sqrt{3}\). The approximate value of \(\sqrt{3}\) is roughly \(1.732\).
Calculating:
\[ 112\sqrt{3} \approx 112 \cdot 1.732 \approx 193.984 \]
Now we check the classification of the number \(112\sqrt{3}\). Since \(\sqrt{3}\) is an irrational number, multiplying it by \(112\) (a rational number) results in an irrational number.
Thus, the product \(112\sqrt{3}\) is an irrational number.
However, the options provided do not match our calculated product of \(193.984\). It is likely that the problem is asking for an approximate value not directly calculated here or an additional step was perhaps indicated.
Regardless, we have confirmed that the product will be an irrational number.
Final answer: \(112\sqrt{3}\), an irrational number.
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