To find the product of \( \sqrt{12} \) and \( 56 \), we can first simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, multiply this by \( 56 \):
\[ 56 \times \sqrt{12} = 56 \times 2\sqrt{3} = 112\sqrt{3} \]
Now, we need to approximate \( \sqrt{3} \) to determine the product's numerical value. The approximate value of \( \sqrt{3} \) is around \( 1.732 \).
Thus:
\[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.184 \]
Now, since \( 112\sqrt{3} \) contains \( \sqrt{3} \), which is irrational, the whole product \( 112\sqrt{3} \) is also irrational.
Given the options you provided, none of them seem to resemble \( 112\sqrt{3} \) directly. However, we know that the product \( 112\sqrt{3} \) is irrational.
So while we can't match it directly to the approximations given, we can conclude that the product is an irrational number.
If you need to select one from the provided options:
- The type of number is irrational, regardless of the specific decimal approximation. The correct answer depends on potentially the closest approximation, but the key takeaway is that the product is an irrational number.
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