To find the product of \( 5 - \sqrt{5} \) and \( \frac{1}{6} \), we perform the multiplication as follows:
\[ \text{Product} = (5 - \sqrt{5}) \cdot \frac{1}{6} \]
Distributing \( \frac{1}{6} \):
\[ \text{Product} = \frac{1}{6} \cdot 5 - \frac{1}{6} \cdot \sqrt{5} \] \[ \text{Product} = \frac{5}{6} - \frac{\sqrt{5}}{6} \]
Now we have the product expressed as:
\[ \frac{5 - \sqrt{5}}{6} \]
Type of Number
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Rationality:
- \( \frac{5}{6} \) is a rational number (since it can be expressed as the ratio of two integers).
- \( \sqrt{5} \) is an irrational number.
- Thus, \( \frac{\sqrt{5}}{6} \) is also irrational.
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Combining Rational and Irrational:
- The expression \( \frac{5 - \sqrt{5}}{6} \) is a combination of a rational number (\( \frac{5}{6} \)) and an irrational part (\( \frac{\sqrt{5}}{6} \)).
- When we subtract an irrational number from a rational number, the result is irrational.
Therefore, the product \( \frac{5 - \sqrt{5}}{6} \) is an irrational number.