To find the product of \( \sqrt{12} \) and \( \frac{5}{6} \), we first simplify \( \sqrt{12} \).
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, we can multiply \( 2\sqrt{3} \) by \( \frac{5}{6} \):
\[ 2\sqrt{3} \times \frac{5}{6} = \frac{2 \times 5 \sqrt{3}}{6} = \frac{10\sqrt{3}}{6} \]
This simplifies to:
\[ \frac{5\sqrt{3}}{3} \]
Now we'll determine the type of number \( \frac{5\sqrt{3}}{3} \) is:
- \( \sqrt{3} \) is an irrational number.
- Therefore, \( 5\sqrt{3} \) is also irrational since the product of a rational number (5) and an irrational number (\(\sqrt{3}\)) is irrational.
- Finally, dividing an irrational number by a rational number (3) still results in an irrational number.
Thus, the product \( \frac{5\sqrt{3}}{3} \) is an irrational number.