To find the product of \(\sqrt{12}\) and \(\frac{5}{6}\), we start by simplifying \(\sqrt{12}\).
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Next, we multiply \(2\sqrt{3}\) by \(\frac{5}{6}\):
\[ 2\sqrt{3} \times \frac{5}{6} = \frac{2 \times 5}{6} \sqrt{3} = \frac{10}{6} \sqrt{3} = \frac{5}{3} \sqrt{3} \]
Now we need to determine the type of number \(\frac{5}{3} \sqrt{3}\) is.
- \(\sqrt{3}\) is an irrational number because it cannot be expressed as a fraction of two integers.
- \(\frac{5}{3}\) is a rational number because it is in the form of a fraction of two integers.
The product of a rational number and an irrational number is irrational. Therefore, \(\frac{5}{3} \sqrt{3}\) is an irrational number.
In summary, the product of \(\sqrt{12}\) and \(\frac{5}{6}\) is \(\frac{5}{3} \sqrt{3}\), which is an irrational number.