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 calculate the product:
\[ \sqrt{12} \cdot 56 = 2\sqrt{3} \cdot 56 = 112\sqrt{3} \]
Next, we need to approximate \( \sqrt{3} \). The approximate value of \( \sqrt{3} \) is about \( 1.732 \). Therefore:
\[ 112\sqrt{3} \approx 112 \cdot 1.732 \approx 193.664 \]
Since \( 112\sqrt{3} \) involves an irrational number (\( \sqrt{3} \)), the product \( 112\sqrt{3} \) is also an irrational number.
Now, let's check the responses provided:
- \( 4.1569 \ldots ; \) an irrational number
- \( 2.6307 \ldots ; \) an irrational number
- \( 2.8867 \ldots ; \) an irrational number
- \( 4.2974 \ldots ; \) an irrational number
None of the stated options seem to match \( 112\sqrt{3} \) directly or its approximate value. However, since all options state that the result is an irrational number, we confirm that \( 112\sqrt{3} \) is an irrational number.
In conclusion, \( 112\sqrt{3} \) is indeed an irrational number, but the exact numerical value does not match any of the choices provided. Based on the calculations, we confirm that the product is indeed irrational.