Question

Find the product of 5–√5 and 1616. What type of number is it?(1 point)

Responses

2.4027 . . . ; an irrational number

2.4027 . . . ; an irrational number

2.40; a rational number

2.40; a rational number

0.37; a rational number

0.37; a rational number

0.3726 . . . ; an irrational number

Answer 1

To find the product of \( 5 - \sqrt{5} \) and \( 1616 \), we first need to calculate \( 5 - \sqrt{5} \) and then multiply it by \( 1616 \).

Calculating \( \sqrt{5} \) approximately gives us:

\[
\sqrt{5} \approx 2.236
\]

Next, \( 5 - \sqrt{5} \) would approximate to:

\[
5 - \sqrt{5} \approx 5 - 2.236 \approx 2.764
\]

Now, we multiply this result by \( 1616 \):

\[
(5 - \sqrt{5}) \times 1616 \approx 2.764 \times 1616
\]

Calculating that product:

\[
2.764 \times 1616 \approx 4476.544
\]

Since we are interested in the exact number rather than an approximation, we have:

\[
(5 - \sqrt{5}) \times 1616 = 1616 \times (5 - \sqrt{5}) = 8080 - 1616\sqrt{5}
\]

Now let's determine the nature of the resulting expression. The term \( \sqrt{5} \) is an irrational number, and multiplying it by any non-zero rational number (like \( 1616 \)) will still yield an irrational number. Therefore, the entire expression \( 8080 - 1616\sqrt{5} \) is also irrational, because it is composed of a rational number \( (8080) \) minus another irrational quantity.

Hence, \( 8080 - 1616\sqrt{5} \) is an **irrational number**.

With the approximate answer checks and from the choices, it seems we would select:

**2.4027...; an irrational number** (if the approximated result corresponds to one of the mentioned responses).