To solve this problem, we'll begin by calculating the product of [tex]\(\sqrt{12}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex].
First, let's find [tex]\(\sqrt{12}\)[/tex]:
[tex]\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}
\][/tex]
Next, we'll multiply [tex]\(2 \sqrt{3}\)[/tex] by [tex]\(\frac{5}{6}\)[/tex]:
[tex]\[
2 \sqrt{3} \times \frac{5}{6} = \frac{2 \sqrt{3} \times 5}{6} = \frac{10 \sqrt{3}}{6} = \frac{5 \sqrt{3}}{3}
\][/tex]
Now, to find the approximate decimal value of [tex]\(\frac{5 \sqrt{3}}{3}\)[/tex]:
[tex]\[
\sqrt{3} \approx 1.732
\][/tex]
[tex]\[
\frac{5 \sqrt{3}}{3} = \frac{5 \times 1.732}{3} = \frac{8.66}{3} \approx 2.8867
\][/tex]
Hence, the product of [tex]\(\sqrt{12}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex] is approximately [tex]\(2.8867\)[/tex].
Finally, we need to determine the type of number this is. Since [tex]\(\sqrt{3}\)[/tex] is an irrational number, any product involving [tex]\(\sqrt{3}\)[/tex] (combined with rational numbers in a non-reducing manner) will also be irrational. Hence, [tex]\(\frac{5 \sqrt{3}}{3}\)[/tex] remains an irrational number.
The correct answer is:
[tex]\(2.8867 \ldots\)[/tex]; an irrational number.
