To find the product of \( \sqrt{12} \) and \( 56 \), we first calculate \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now, we can calculate the product:
\[ \sqrt{12} \cdot 56 = 2\sqrt{3} \cdot 56 = 112\sqrt{3} \]
Next, we need to evaluate \( 112\sqrt{3} \) to find its approximate value. The value of \( \sqrt{3} \) is approximately \( 1.732 \). Thus, we can compute:
\[ 112\sqrt{3} \approx 112 \cdot 1.732 \approx 193.824 \]
Since \( \sqrt{3} \) is an irrational number, any product involving \( \sqrt{3} \) (like \( 112\sqrt{3} \)) will also be an irrational number.
Now looking at the provided options:
- 2.6307...; an irrational number
- 4.1569...; an irrational number
- 4.2974...; an irrational number
- 2.8867...; an irrational number
None of these options reflect the value we calculated. However, we do know \( 112\sqrt{3} \) is an irrational number.
If you need to check the accuracy of the options, you can compute \( 112\sqrt{3} \) more rigorously, but unfortunately, none reflect the anticipated product correctly.
Thus, the correct answer is that \( 112\sqrt{3} \) is an irrational number, but the specific numeric approximation does not match any of your current choices.