Find the product of 5–√

Question

and 1/6
. What type of number is it?(1 point)
Responses

0.37; a rational number
0.37; a rational number

0.3726 . . . ; an irrational number
0.3726 . . . ; an irrational number

2.40; a rational number
2.40; a rational number

2.4027 . . . ; an irrational number

Answer 1

To find the product of \(5 - \sqrt{5}\) and \(\frac{1}{6}\), we first need to calculate \(5 - \sqrt{5}\).

1. The square root of 5, \(\sqrt{5}\), is approximately \(2.236\).
2. Therefore, \(5 - \sqrt{5} \approx 5 - 2.236 = 2.764\).
3. Now, we multiply this by \(\frac{1}{6}\): \[ (5 - \sqrt{5}) \cdot \frac{1}{6} \approx 2.764 \cdot \frac{1}{6} \approx 0.46067 \]

Now, let's express the product exactly:

\[ \frac{5 - \sqrt{5}}{6} \]

Next, we need to determine whether this number is rational or irrational. The term \(5 - \sqrt{5}\) combines a rational number (5) with an irrational number (\(\sqrt{5}\)), which results in an irrational number. Dividing an irrational number by a ration (like 6) still results in an irrational number.

Thus, \(\frac{5 - \sqrt{5}}{6}\) is an irrational number.

Based on the possible responses, the closest choice is:

0.3726 . . . ; an irrational number.