To find the product of \( 5 - \sqrt{2} \) and \( 16 \), we calculate:
\[ (5 - \sqrt{2}) \times 16 = 80 - 16\sqrt{2} \]
Now, to determine the type of number \( 80 - 16\sqrt{2} \) is, we know that \( \sqrt{2} \) is an irrational number. Therefore, \( 16\sqrt{2} \) is also an irrational number. The difference between a rational number (80, which is rational) and an irrational number (\( 16\sqrt{2} \)) is always irrational.
Thus, \( 80 - 16\sqrt{2} \) is an irrational number.
Based on the responses provided, the correct response would be:
0.3726 . . . ; an irrational number (noting that this is not the actual value of the product but describes the type).
However, it seems there might be a confusion in the responses you provided since the specific values don't match up. The key takeaway is that \( 80 - 16\sqrt{2} \) is indeed an irrational number.
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