To find the product of \( \sqrt{12} \) and \( 56 \), we first calculate \( \sqrt{12} \).
We know: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now we compute the product: \[ \sqrt{12} \times 56 = 2\sqrt{3} \times 56 = 112\sqrt{3} \]
Next, we need to determine the type of number \( 112\sqrt{3} \). Since \( \sqrt{3} \) is an irrational number, and any non-zero rational number multiplied by an irrational number results in an irrational number, \( 112\sqrt{3} \) is also an irrational number.
Now, for a numerical approximation of \( 112\sqrt{3} \): \[ \sqrt{3} \approx 1.732 \] Thus, \[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.184 \]
The product \( 112\sqrt{3} \) is irrational.
None of the provided responses match the calculated product, but we can affirm that:
- The product \( 112\sqrt{3} \) is an irrational number.
Therefore, the conclusion is:
- The product is an irrational number. (Knowing the numerical values did not lead to a match from given options.)