Question

Find the product of 5–√
and 16
. What type of number is it?(1 point)
Responses

0.37; a rational number
0.37; a rational number

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

Answer 1

To find the product of \( 5 - \sqrt{5} \) and \( 16 \), we can calculate it as follows:

\[
(5 - \sqrt{5}) \times 16 = 16(5 - \sqrt{5})
\]

Expanding that, we have:

\[
16 \times 5 - 16 \times \sqrt{5} = 80 - 16\sqrt{5}
\]

Now, \( 80 \) is a rational number and \( 16\sqrt{5} \) involves \( \sqrt{5} \), which is an irrational number. Therefore, \( 80 - 16\sqrt{5} \) will also be an irrational number because subtracting a rational number from an irrational number results in an irrational number.

Thus, the product is \( 80 - 16\sqrt{5} \), which is an irrational number.

Looking at the provided responses:

- 0.37; a rational number
- 2.4027 . . . ; an irrational number
- 0.3726 . . . ; an irrational number
- 2.40; a rational number

None of these options match our calculated result directly, but we correctly identified that the product is an irrational number. The response "2.4027 . . . ; an irrational number" is closest in form (as it suggests an irrational number), even though the exact numerical match might not be correct based on our calculation.

Thus, the closest correct response to select would be:

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