To find the product of \(5 - \sqrt{5}\) and \(16\), we can perform the multiplication as follows:
\[ 16 \times (5 - \sqrt{5}) = 16 \times 5 - 16 \times \sqrt{5} \]
Calculating \(16 \times 5\):
\[ 16 \times 5 = 80 \]
Calculating \(16 \times \sqrt{5}\):
\[ 16 \times \sqrt{5} = 16\sqrt{5} \]
So the expression becomes:
\[ 80 - 16\sqrt{5} \]
Now, we need to analyze the nature of this number.
- The number \(80\) is a rational number (as it can be expressed as \(\frac{80}{1}\)).
- The term \(16\sqrt{5}\) involves \(\sqrt{5}\), which is an irrational number.
Subtracting an irrational number \(16\sqrt{5}\) from a rational number \(80\) results in an irrational number, as the set of rational numbers is closed under subtraction, but this operation involving an irrational number leads us beyond the set of rational numbers.
Therefore, \(80 - 16\sqrt{5}\) is an irrational number.
As a result, the final answer is:
\(80 - 16\sqrt{5}\); an irrational number. Since that doesn't match your options, the best conclusion is that the product \(2.4027...\) will represent an approximate value of \(80 - 16\sqrt{5}\), and it is indeed an irrational number.
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