If n is a natural number, then `sqrtn` is

Options

A. always a rational number
B. always a natural number
C. always an irrational number
D. sometimes a natural number and sometimes an irrational number

Answer

To solve the problem, we need to analyze the expression \(\sqrt{n}\) where \(n\) is a natural number.

### Step-by-Step Solution:

1. **Understanding Natural Numbers**:
- Natural numbers are the set of positive integers starting from 1, i.e., \(1, 2, 3, 4, 5, \ldots\).

2. **Evaluating \(\sqrt{n}\)**:
- The square root of a natural number \(n\) can either be a natural number or an irrational number.
- For example:
- If \(n = 1\), then \(\sqrt{1} = 1\) (natural number).
- If \(n = 4\), then \(\sqrt{4} = 2\) (natural number).
- If \(n = 2\), then \(\sqrt{2}\) is an irrational number.
- If \(n = 3\), then \(\sqrt{3}\) is also an irrational number.

3. **Categorizing Natural Numbers**:
- We can categorize natural numbers into two groups:
- **Perfect Squares**: Numbers like \(1, 4, 9, 16, \ldots\) which have integer square roots.
- **Non-Perfect Squares**: Numbers like \(2, 3, 5, 6, 7, 8, \ldots\) which have irrational square roots.

4. **Conclusion**:
- Therefore, if \(n\) is a natural number:
- \(\sqrt{n}\) is a natural number if \(n\) is a perfect square.
- \(\sqrt{n}\) is an irrational number if \(n\) is not a perfect square.
- Hence, we can conclude that \(\sqrt{n}\) may sometimes be a natural number or sometimes an irrational number.

### Final Answer:
If \(n\) is a natural number, then \(\sqrt{n}\) is sometimes a natural number and sometimes an irrational number.