To find the product of \( 5 - \sqrt{5} \) and \( \frac{1}{6} \), we first calculate the value of \( 5 - \sqrt{5} \).
- First, estimate \( \sqrt{5} \). The square root of 5 is approximately 2.236.
So, \( 5 - \sqrt{5} \approx 5 - 2.236 \approx 2.764 \).
- Now, multiply this result by \( \frac{1}{6} \):
\[ (5 - \sqrt{5}) \cdot \frac{1}{6} \approx 2.764 \cdot \frac{1}{6} \approx \frac{2.764}{6} \approx 0.46067 \]
However, it is illustrative to calculate it without estimating if we want to consider the exact nature of the number.
- The exact calculation is:
\[ 5 - \sqrt{5} \text{ is a real number.} \]
When we multiply it with \( \frac{1}{6} \),
\[ \text{The product is} \frac{5 - \sqrt{5}}{6}. \]
This expression is still a real number. The \( \sqrt{5} \) is an irrational number, and \( 5 - \sqrt{5} \) is also irrational. When multiplying an irrational number by a rational number (like \( \frac{1}{6} \)), the result is an irrational number.
Now, from the options given, we can summarize that:
- The result \( \frac{5 - \sqrt{5}}{6} \) is approximately 0.46067, which isn't listed explicitly in the options.
However, its classification as an irrational number is.
Conclusion:
Since the number is determined to be irrational, but does not match the approximations given precisely, we can say the result from the above calculations confirms it is of type irrational.
Rationale:
- Product: The product is approximately \( 0.46067 \) and is an irrational number.
Thus the appropriate answer closest to your options is:
0.3726 . . . ; an irrational number (if "approximation" is to be considered for options given) or 2.4027 . . . ; an irrational number (if referencing more than one decimal).