To find the product of \( 5 - \sqrt{5} \) and \( 16 \), we perform the multiplication:
\[ 16(5 - \sqrt{5}) = 16 \cdot 5 - 16 \cdot \sqrt{5} = 80 - 16\sqrt{5}. \]
Next, we need to determine the type of number \( 80 - 16\sqrt{5} \) is.
Since \( \sqrt{5} \) is an irrational number, any non-zero rational number multiplied by an irrational number (in this case, \( -16\sqrt{5} \)), will yield an irrational number. Therefore, \( 80 - 16\sqrt{5} \) is also irrational because it combines a rational number \( 80 \) and the irrational part \( -16\sqrt{5} \).
Thus, the answer would be expressed as:
\[ 80 - 16\sqrt{5}; \text{ an irrational number}. \]
However, looking at the options provided, none of them appear to match our final expression exactly.
If we wanted to find a numerical approximation for \( 80 - 16\sqrt{5} \):
- Calculate \( \sqrt{5} \) (approximately \( 2.236 \)).
- Then \( 16\sqrt{5} \approx 16 \times 2.236 = 35.776 \).
- Finally, \( 80 - 35.776 \approx 44.224. \)
Since the approximation is not directly provided, it seems like the answer is primarily focusing on whether the resultant number is rational or irrational, with the conclusion that it is indeed irrational.
Given the options, we conclude that none fit our calculation precisely, but the process led us to determine that \( 80 - 16\sqrt{5} \) is an irrational number.
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