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).