To find the product of \(\sqrt{12}\) and \(\frac{5}{6}\), we start by simplifying \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, we can multiply \(2\sqrt{3}\) by \(\frac{5}{6}\):
\[ 2\sqrt{3} \times \frac{5}{6} = \frac{2 \times 5}{6} \times \sqrt{3} = \frac{10}{6} \sqrt{3} = \frac{5}{3} \sqrt{3} \]
Now let's analyze the number \(\frac{5}{3} \sqrt{3}\):
- \(\sqrt{3}\) is an irrational number, meaning it cannot be expressed as a fraction.
- When you multiply a rational number (\(\frac{5}{3}\)) by an irrational number (\(\sqrt{3}\)), the result is also an irrational number.
Thus, the product of \(\sqrt{12}\) and \(\frac{5}{6}\), which is \(\frac{5}{3} \sqrt{3}\), is an irrational number.