To find the product of \( \sqrt{12} \) and \( 56 \), we first need to calculate \( \sqrt{12} \).
We know that: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now, let's calculate the product: \[ \sqrt{12} \times 56 = (2\sqrt{3}) \times 56 = 112\sqrt{3} \]
Next, we need to approximate \( \sqrt{3} \). The approximate value of \( \sqrt{3} \) is about \( 1.732 \), so: \[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.984 \]
Since \( 112\sqrt{3} \) itself is in exact form, we turn it into a decimal approximation by computing: \[ 112 \times 1.732 = 193.984 \]
Now, the number \( 193.984 \) is definitely not one of the options you listed.
Looking closely at the options listed, it appears they concern only decimal approximations resulting from multiplying a reduced or simplified version of the initial expression. However, most importantly, \( 112\sqrt{3} \) is an irrational number since it contains the irrational part \( \sqrt{3} \).
In light of this, none of the options are correct as they do not correlate with the calculation of \( \sqrt{12} \times 56 \).
To clarify the nature of \( 112\sqrt{3} \):
- The product \( 112\sqrt{3} \) is an irrational number.
Given the calculations above, the product itself is \( 112\sqrt{3} \), and it is classified as an irrational number.
If there was some miscommunication about which value to use for \( 12^{0.5} \) and \( 56 \), kindly provide further instructions, or if there's an alternative method to the initial query. Please note, though, that \( 112\sqrt{3} \) approximates \( 193.984 \) and is indeed an irrational number.
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