To find the product of \( \sqrt{12} \) and \( 56 \), we can calculate it as follows:
First, simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now, multiply by \( 56 \):
\[ 56 \cdot \sqrt{12} = 56 \cdot 2\sqrt{3} = 112\sqrt{3} \]
Next, we approximate the value of \( \sqrt{3} \):
\[ \sqrt{3} \approx 1.732 \]
Therefore:
\[ 112\sqrt{3} \approx 112 \cdot 1.732 \approx 193.184 \]
Analyzing the options provided, \( 112\sqrt{3} \) is an irrational number because it involves \( \sqrt{3} \), which is irrational.
None of the numeric values provided (4.1569, 2.6307, 4.2974, 2.8867) appear to align with our calculated approximate value, so it is important to note that the product explicitly remains as \( 112\sqrt{3} \) and it is confirmed to be an irrational number.
Thus, the final result is:
112√3; an irrational number.