The problem asks us to find the product of $\sqrt{12}$ and $3\sqrt{27}$ in simplest form. We also need to determine if the result is rational or irrational and explain why.

1. Problem Description

The problem asks us to find the product of $\sqrt{12}$ and $3\sqrt{27}$ in simplest form. We also need to determine if the result is rational or irrational and explain why.

2. Solution Steps

First, we simplify each radical:

$\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$

$3\sqrt{27} = 3\sqrt{9 \cdot 3} = 3(\sqrt{9} \cdot \sqrt{3}) = 3(3\sqrt{3}) = 9\sqrt{3}$

Now, we multiply the simplified radicals:

$(2\sqrt{3})(9\sqrt{3}) = 2 \cdot 9 \cdot \sqrt{3} \cdot \sqrt{3} = 18 \cdot 3 = 54$

Since $54$ can be written as a fraction $\frac{54}{1}$, it is a rational number. A rational number can be expressed as a ratio of two integers. The decimal expansion of a rational number either terminates or repeats. In this case, the decimal expansion terminates (54.0).

3. Final Answer

Result: 54

The result is rational because it can be written as the ratio of two integers and its decimal expansion terminates.