To find the product of \( \sqrt{12} \) and \( 56 \), we first calculate \( \sqrt{12} \).
We know that: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now, we can multiply this by \( 56 \): \[ 56 \cdot \sqrt{12} = 56 \cdot 2\sqrt{3} = 112\sqrt{3} \]
Next, we calculate the approximate value of \( 112\sqrt{3} \). The approximate value of \( \sqrt{3} \) is about \( 1.732 \). Thus, \[ 112\sqrt{3} \approx 112 \cdot 1.732 \approx 193.824 \]
As \( \sqrt{3} \) is an irrational number, the product \( 112\sqrt{3} \) is also irrational.
Given the above information, the product is approximately \( 193.824 \) which does not match the provided options in terms of specific numerical values; however, all options indicate that the output is an irrational number.
Therefore, the product \( 112\sqrt{3} \) is an irrational number.
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