To find the product of \( \sqrt{12} \) and \( 56 \), we first calculate \( \sqrt{12} \).
We know that:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, we can multiply this by \( 56 \):
\[ 56 \cdot \sqrt{12} = 56 \cdot 2\sqrt{3} = 112\sqrt{3} \]
Next, we should approximate \( \sqrt{3} \) to find the numerical value of \( 112 \sqrt{3} \).
The approximate value of \( \sqrt{3} \) is about \( 1.732 \). Thus, we can calculate:
\[ 112 \cdot 1.732 \approx 193.184 \]
To determine the exact nature of the number \( 112\sqrt{3} \):
- Since \( \sqrt{3} \) is an irrational number, and the product of a rational number (in this case, \( 112 \)) and an irrational number is also irrational, we can conclude that \( 112\sqrt{3} \) is irrational.
However, this doesn't match directly with any of the response values given. Let's assume these responses are results of approximating or calculating \( \sqrt{12} \cdot 56 \), let's evaluate:
Calculating \( 56 \cdot \sqrt{12} \) numerically with \( \sqrt{12} \approx 3.464 \):
\[ 56 \cdot 3.464 \approx 193.984 \]
But among the available responses, we compare:
- \( 4.1569 \), \( 2.6307 \), \( 2.8867 \), and \( 4.2974 \) all appear incorrect for approximating \( 112\sqrt{3} = 193.984 \).
It appears there was a misalignment in either the selected values or expectations from calculations. The final nature of the product \( 112\sqrt{3} \) is definitively an irrational number, as reasoned, but with no exact match to numerical results expected in your options.
If I were to answer based on your options, I might assert that they collectively imply all output irrational figures, but none concretely reflects the computed product.
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