To find the product of \(\sqrt{12}\) and \(\frac{5}{6}\), we first calculate \(\sqrt{12}\).
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Next, we need to multiply this by \(\frac{5}{6}\):
\[ \text{Product} = 2\sqrt{3} \times \frac{5}{6} = \frac{2 \times 5 \sqrt{3}}{6} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3} \]
Now, to determine whether \(\frac{5\sqrt{3}}{3}\) is rational or irrational, we note that \(\sqrt{3}\) is an irrational number (since 3 is not a perfect square). The product of a rational number (\(\frac{5}{3}\)) and an irrational number (\(\sqrt{3}\)) is also irrational.
Thus, the product we calculated is \(\frac{5\sqrt{3}}{3}\), which is an irrational number.
Now, let's approximate \(\sqrt{3}\) to find the numerical value of the product:
\[ \sqrt{3} \approx 1.732 \]
Calculating \(\frac{5}{3} \cdot 1.732\):
\[ \frac{5}{3} \approx 1.6667 \]
So:
\[ 1.6667 \cdot 1.732 \approx 2.8867 \]
Putting it all together, we find that the product is approximately \(2.8867\).
The final response is:
2.8867 . . . ; an irrational number.
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