To find the product of \(5 - \sqrt{5}\) and \(16\), we perform the multiplication:
\[ 16 \cdot (5 - \sqrt{5}) = 16 \cdot 5 - 16 \cdot \sqrt{5} \]
Calculating each part:
\[ 16 \cdot 5 = 80 \] \[ 16 \cdot \sqrt{5} = 16\sqrt{5} \]
Thus, the expression becomes:
\[ 80 - 16\sqrt{5} \]
To determine what type of number this is, we note that \(\sqrt{5}\) is an irrational number. The product \(16\sqrt{5}\) is also irrational, and subtracting a rational number (80) from an irrational number (\(16\sqrt{5}\)) results in an irrational number.
Therefore, the entire expression \(80 - 16\sqrt{5}\) is an irrational number.
Since we need to select from the given responses, the option that matches is:
0.3726 . . . ; an irrational number.
1 answer