To analyze \(\sqrt{255}\), we first need to factor the number 255 into its prime factors.
- Find the prime factorization of 255:
- Since 255 is an odd number, it is not divisible by 2.
- Check divisibility by 3: \(2 + 5 + 5 = 12\), which is divisible by 3. Therefore, \(255 \div 3 = 85\).
- Now, factor 85: it is also odd and not divisible by 3. Check for 5 (the last digit is 5): \(85 \div 5 = 17\).
- Lastly, 17 is a prime number.
So, the prime factorization of 255 is:
\[ 255 = 3 \times 5 \times 17 \]
- To find \(\sqrt{255}\), we can express it using its prime factors:
\[ \sqrt{255} = \sqrt{3 \times 5 \times 17} \]
- This can be simplified further to:
\[ \sqrt{255} = \sqrt{3} \times \sqrt{5} \times \sqrt{17} \]
Since none of the prime factors are perfect squares, \(\sqrt{255}\) cannot be simplified further into a rational number, and it remains an irrational number.
Conclusion:
The factors of 255 are \(3, 5,\) and \(17\), and \(\sqrt{255}\) can be expressed as \(\sqrt{3} \times \sqrt{5} \times \sqrt{17}\), indicating that \(\sqrt{255}\) is irrational.