Question

(√8 , √9 , √10 ) rational only, irrational only, or both?

Answer 1

To determine whether the numbers \( \sqrt{8} \), \( \sqrt{9} \), and \( \sqrt{10} \) are rational, irrational, or both, we can evaluate each square root individually:

1. **\( \sqrt{8} \)**:
- Simplifying \( \sqrt{8} \):
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\]
- Since \( \sqrt{2} \) is an irrational number, \( 2\sqrt{2} \) is also irrational.
- Therefore, \( \sqrt{8} \) is **irrational**.

2. **\( \sqrt{9} \)**:
- Simplifying \( \sqrt{9} \):
\[
\sqrt{9} = 3
\]
- Since \( 3 \) is a whole number, it is a rational number.
- Therefore, \( \sqrt{9} \) is **rational**.

3. **\( \sqrt{10} \)**:
- The number \( 10 \) is not a perfect square, and thus \( \sqrt{10} \) cannot be simplified to a rational number.
- Therefore, \( \sqrt{10} \) is **irrational**.

In summary:
- \( \sqrt{8} \): irrational
- \( \sqrt{9} \): rational
- \( \sqrt{10} \): irrational

So the overall classification is:
- **Rational only**: \( \sqrt{9} \)
- **Irrational only**: \( \sqrt{8} \) and \( \sqrt{10} \)
- **Both**: None