To find the product of \( \sqrt{12} \) and \( 56 \), we first calculate the exact value of \( \sqrt{12} \).
\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now we calculate the product with \( 56 \):
\[ 56 \cdot \sqrt{12} = 56 \cdot 2\sqrt{3} = 112\sqrt{3} \]
Next, we need to evaluate \( 112\sqrt{3} \) numerically. The approximate value of \( \sqrt{3} \) is \( 1.732 \).
Now we calculate \( 112 \cdot 1.732 \):
\[ 112 \cdot 1.732 \approx 193.184 \]
To narrow down the answer, we need to find the approximate decimal value.
Given that \( \sqrt{3} \approx 1.732 \),
\[ 112 \cdot 1.732 \approx 193.184 \]
The result of \( 112\sqrt{3} \) is an irrational number because \( \sqrt{3} \) is irrational.
Now checking the options provided, none of them match the previous estimate of \( 193.184 \).
To find a more specific option among the given multiple-choice responses regarding the nature of the number:
- All responses state the result is an irrational number.
- Therefore, we only need to match the approximate numerical values given in the options.
Now let's compute \( 112 \cdot \sqrt{3} \) again to see if any of the decimal approximations falls approximately to one of the given responses:
Using \( \sqrt{3} \approx 1.73205 \):
Calculating:
\[ 112 \cdot 1.73205 \approx 193.1856 \]
Verifying with values:
It appears that there was some miscommunication regarding the nature of the options. However, since all responses confirm the product as an irrational number, and exactly computing \( 56 \cdot \sqrt{12} = 112\sqrt{3} \) confirms it being irrational.
Thus, confirming the nature of the calculation as follows: \( 112\sqrt{3} \) is indeed irrational.
Among the options given, none seem to directly match. Assuming they are targeting values of a product near a certain rounding, they are more about approximative estimates rather than \( 112\sqrt{3} \).
In conclusion, choosing an irrational number seems valid but requires critical verification of the value. If we compare it against provided estimates, depending on leveling their computation ranges, they all produce coherent estimation for \( 112\sqrt{3} \) being irrational, leading to confirming it's indeed an irrational number without exact alignment on numeric responses listed in choices presented.
Final Answer: \( 112\sqrt{3} \); an irrational number.
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