To find the product of \( \sqrt{12} \) and \( 56 \), we start by calculating \( \sqrt{12} \).
First, simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Next, calculate the product:
\[ \sqrt{12} \times 56 = 2\sqrt{3} \times 56 = 112\sqrt{3} \]
Now, we need to find \( 112\sqrt{3} \) numerically. Since \( \sqrt{3} \approx 1.732 \):
\[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.184 \]
Since \( \sqrt{3} \) is an irrational number, \( 112\sqrt{3} \) is also an irrational number because the product of a rational number (112) and an irrational number (\(\sqrt{3}\)) is irrational.
Now, let’s assess the options provided to find a matching decimal:
To find the decimal of \( 112\sqrt{3} \), we compute:
\[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.184 \]
Since this differs from the options you provided, it seems there may have been a misunderstanding in interpreting the question or the options themselves.
However, since \( 112\sqrt{3} \) is indeed an irrational number, the key takeaway is that the product \( \sqrt{12} \times 56 \) is irrational.
If you're looking for a specific value that matches one of the options given (2.8867, 2.6307, 4.1569, or 4.2974), none of these correspond to \( 112\sqrt{3} \). Please check the numbers you provided.
The product \( 112\sqrt{3} \) is rational, approximately equal to 193.184, and is indeed an irrational number.