To find the product of \( \sqrt{12} \) and \( 56 \), we can start by calculating \( \sqrt{12} \).
We know that: \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}. \]
Now, we can calculate the product: \[ \sqrt{12} \times 56 = 2\sqrt{3} \times 56 = 112\sqrt{3}. \]
Next, we can estimate the value of \( \sqrt{3} \): \[ \sqrt{3} \approx 1.732. \]
Therefore, we can find \( 112\sqrt{3} \) by multiplying: \[ 112 \times 1.732 \approx 193.184. \]
Since \( 112\sqrt{3} \) is a product of a rational number (112) and an irrational number (\( \sqrt{3} \)), the result \( 112\sqrt{3} \) is also irrational.
Looking at your provided responses, none of them list \( 193.184 \) as an option. However, since \( 112\sqrt{3} \) is irrational, it confirms that the product is an irrational number. Without an exact match in the product value, the most pertinent information is that the product is:
Answer: \( 112\sqrt{3} \); an irrational number.
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