To find the product of \( \sqrt{12} \) and \( \frac{5}{6} \), we first calculate \( \sqrt{12} \).
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Now, we multiply this result by \( \frac{5}{6} \):
\[ \text{Product} = \frac{5}{6} \cdot 2\sqrt{3} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3} \]
Next, we can approximate \( \sqrt{3} \). We know \( \sqrt{3} \approx 1.732 \):
\[ \text{Approximation} = \frac{5 \cdot 1.732}{3} \approx \frac{8.66}{3} \approx 2.8867 \]
Thus, \( \frac{5\sqrt{3}}{3} \approx 2.8867 \).
Since \( \sqrt{3} \) is an irrational number, and multiplying it by a rational number (\(\frac{5}{3}\)) still results in an irrational number, we conclude:
The product is approximately \( 2.8867 \ldots \); it is an irrational number.
Therefore, the correct response is:
2.8867 . . . ; an irrational number.
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