To find the product of \( \sqrt{12} \) and \( 56 \), we can calculate it step by step.
First, evaluate \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, multiply it by \( 56 \): \[ 56 \times \sqrt{12} = 56 \times 2\sqrt{3} = 112\sqrt{3} \]
Next, we need to evaluate \( \sqrt{3} \) using its approximate value, which is approximately \( 1.732 \): \[ 112 \times \sqrt{3} \approx 112 \times 1.732 \approx 193.184 \]
This number, \( 112\sqrt{3} \), falls within the realm of irrational numbers because \( \sqrt{3} \) is irrational and the product of a non-zero rational number and an irrational number is also irrational.
Among the choices provided, we can approximate \( 112\sqrt{3} \) and determine the type:
Thus, as it stands, \( 112\sqrt{3} \) is an irrational number, but the exact decimal approximations are not provided directly from the given choices. However, I can confirm that the product \( 112\sqrt{3} \) is indeed irrational.
The options given seem to contain some numerical values that suggest approximations of different irrational products. Generally, we would select the one that correctly represents the product \( 56\sqrt{12} \) or \( 112\sqrt{3} \) and is stated to be irrational.
Please confirm if you need a specific numerical approximation that much closely resembles \( 112\sqrt{3} \). Otherwise, as stated, the product is an irrational number.