Question

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

2.40; a rational number
2.40; a rational number

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

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

0.37; a rational number

Answer 1

To find the product of \(5 - \sqrt{5}\) and \(16\), we perform the multiplication:

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

Calculating each part:

\[
16 \cdot 5 = 80
\]
\[
16 \cdot \sqrt{5} = 16\sqrt{5}
\]

Thus, the expression becomes:

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

To determine what type of number this is, we note that \(\sqrt{5}\) is an irrational number. The product \(16\sqrt{5}\) is also irrational, and subtracting a rational number (80) from an irrational number (\(16\sqrt{5}\)) results in an irrational number.

Therefore, the entire expression \(80 - 16\sqrt{5}\) is an irrational number.

Since we need to select from the given responses, the option that matches is:

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