Question

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

2.4027 . . . ; an irrational number
2.4027 . . . ; an irrational number

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

2.40; a rational number
2.40; a rational number

0.37; a rational number
0.37; a rational number

Answer 1

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

Calculating \( 5 - \sqrt{5} \):
- The approximate value of \( \sqrt{5} \) is about 2.236.
- Therefore, \( 5 - \sqrt{5} \approx 5 - 2.236 = 2.764 \).

Now, we will find the product:

\[
(5 - \sqrt{5}) \cdot \frac{1}{6} = \frac{5 - \sqrt{5}}{6}
\]

Using the approximate value we calculated:

\[
\frac{2.764}{6} \approx 0.4607
\]

Now, let's evaluate whether this value is rational or irrational:

- The number \( 5 - \sqrt{5} \) is irrational (since \( \sqrt{5} \) is irrational, and the subtraction of a rational number from an irrational number remains irrational).
- Multiplying an irrational number (\( 5 - \sqrt{5} \)) by a rational number (\( \frac{1}{6} \)) results in an irrational number.

Thus, approximately \( 0.4607 \) is an irrational number.

However, the response choices do not include \( 0.4607; an irrational number \). Provided choices should be evaluated based on their closeness to \( \frac{5 - \sqrt{5}}{6} \):

- The closest choice is \( 0.3726 \ldots; an irrational number \) or \( 0.37; a rational number \).

The correct classification must indicate that the product is indeed irrational, but no exact computation confirms \( 0.3726 \ldots \) from \( \frac{5 - \sqrt{5}}{6} \).

However, selecting options relative to the product signify:
- **0.3726 . . . ; an irrational number** seems more suitable due to irrationality.

So the final answer is:

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