Let's calculate the product of \( \sqrt{12} \) and \( 56 \):
First, we know that \( \sqrt{12} \) can be simplified: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now, we compute the product: \[ \sqrt{12} \cdot 56 = 2\sqrt{3} \cdot 56 = 112\sqrt{3} \]
Next, let's estimate the value of \( \sqrt{3} \). The approximate value of \( \sqrt{3} \) is about \( 1.732 \).
Now we calculate: \[ 112 \cdot 1.732 \approx 193.984 \]
Since \( \sqrt{3} \) is an irrational number, \( 112\sqrt{3} \) is also irrational.
Thus, the product \( \sqrt{12} \cdot 56 = 112\sqrt{3} \) is an irrational number.
Among the response options provided, the correct answer would be:
- 4.1569 . . . ; an irrational number
It seems the numerical value provided does not match our calculation (it is not exactly \( 4.1569 \)), but the type we concluded is indeed an irrational number.