Question

Find the product of √ 12and 5/6. What type of number is it?(1 point)
Responses

4.1569 . . . ; an irrational number

2.6307 . . . ; an irrational number

2.8867 . . . ; an irrational number

4.2974 . . . ; an irrational number

Answer 1

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**.