To find the product of \( \sqrt{12} \) and \( \frac{5}{6} \), we first compute \( \sqrt{12} \).
We know that: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Next, we multiply \( 2\sqrt{3} \) by \( \frac{5}{6} \): \[ 2\sqrt{3} \times \frac{5}{6} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3} \]
Now, we need to compute the approximate value of \( \frac{5\sqrt{3}}{3} \). Given that \( \sqrt{3} \approx 1.732 \): \[ \frac{5 \times 1.732}{3} = \frac{8.66}{3} \approx 2.8867 \]
Thus, the product \( \frac{5\sqrt{3}}{3} \approx 2.8867 \).
Since \( \sqrt{3} \) is an irrational number, any product of an irrational number and a non-zero rational number is also irrational.
The final result is: \[ 2.8867... ; \text{ an irrational number} \]
Therefore, the correct response is: 2.8867 . . . ; an irrational number
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