To find the product of \( 5 - \sqrt{5} \) and \( 16 \), we first need to calculate it:
\[ 16 \times (5 - \sqrt{5}) = 16 \times 5 - 16 \times \sqrt{5} \]
Calculating each term:
\[ 16 \times 5 = 80 \] \[ 16 \times \sqrt{5} \]
As such, the product becomes:
\[ 80 - 16\sqrt{5} \]
Next, we need to determine the nature of the resulting number:
- \( 80 \) is a rational number.
- \( 16\sqrt{5} \) is irrational because the square root of 5 is irrational, and multiplying an irrational number by a rational number results in an irrational number.
The expression \( 80 - 16\sqrt{5} \) consists of a rational part \( 80 \) and an irrational part \( -16\sqrt{5} \). The sum (or difference) of a rational and an irrational number is always irrational.
Therefore, \( 80 - 16\sqrt{5} \) is an irrational number.
Looking at the options provided, none of the numbers listed directly match this expression. However, it is confirmed that the product is an irrational number.
So, the final answer is that the product is an irrational number.
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