To find the product of \( \sqrt{12} \) and \( \frac{5}{6} \):
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First, calculate \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
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Now, multiply \( 2\sqrt{3} \) by \( \frac{5}{6} \): \[ 2\sqrt{3} \times \frac{5}{6} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3} \]
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Next, we can approximate \( \sqrt{3} \): \[ \sqrt{3} \approx 1.732 \] So, \[ \frac{5\sqrt{3}}{3} \approx \frac{5 \times 1.732}{3} \approx \frac{8.66}{3} \approx 2.8867 \]
Now we consider the nature of the number \( 2.8867 \). Since \( \sqrt{3} \) is an irrational number, \( \frac{5\sqrt{3}}{3} \) is also irrational (as the product and division of rational numbers and irrational numbers remain irrational).
Thus, the product of \( \sqrt{12} \) and \( \frac{5}{6} \) is approximately \( 2.8867 \) and is an irrational number.
The correct answer is: 2.8867 an irrational number.