Question

Let
be a nonzero rational number. Is

rational or irrational?


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Answer 1

To determine whether \( \sqrt{x} \) is rational or irrational when \( x \) is a nonzero rational number, we follow these steps:

1. **Definition**: A rational number is defined as a number that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

2. **Consider \( x \)**: Let \( x = \frac{p}{q} \), where \( p \) and \( q \) are integers (with \( q \neq 0 \)).

3. **Square Root**: We need to evaluate \( \sqrt{x} = \sqrt{\frac{p}{q}} \).

4. **Simplification**: We can use the property of square roots:
\[
\sqrt{\frac{p}{q}} = \frac{\sqrt{p}}{\sqrt{q}}.
\]

5. **Analysis**: Now, we analyze \( \sqrt{p} \) and \( \sqrt{q} \):
- If both \( p \) and \( q \) are perfect squares, then \( \sqrt{p} \) and \( \sqrt{q} \) are both rational, meaning \( \sqrt{x} \) is rational.
- If either \( p \) or \( q \) is not a perfect square, then \( \sqrt{p} \) or \( \sqrt{q} \) will be irrational, making \( \sqrt{x} \) irrational.

6. **Conclusion**: Thus, \( \sqrt{x} \) is rational if and only if both \( p \) and \( q \) are perfect squares.

**Final Answer**: \( \sqrt{x} \) can be rational or irrational, depending on the specific values of \( p \) and \( q \).