Question

Which of the following options includes a rational number and an irrational number between √6 and √8 ?(1 point)Responses
2.6¯, √7
2.5, π
9/4, √7
2.7, √9

Answer 1

To identify which option includes a rational number and an irrational number between \(\sqrt{6}\) and \(\sqrt{8}\), let's first calculate the approximate values of these square roots.

- \(\sqrt{6} \approx 2.45\)
- \(\sqrt{8} \approx 2.83\)

Now we can analyze the options provided:

1. **\(2.6\bar{6}, \sqrt{7}\)**:
- \(2.6\bar{6} \approx 2.666...\) (rational, as it can be expressed as a fraction)
- \(\sqrt{7} \approx 2.64575...\) (irrational)
- Both numbers are between \(\sqrt{6}\) and \(\sqrt{8}\).

2. **\(2.5, \pi\)**:
- \(2.5\) (rational)
- \(\pi \approx 3.14\) (irrational and not between \(\sqrt{6}\) and \(\sqrt{8}\))

3. **\(9/4, \sqrt{7}\)**:
- \(9/4 = 2.25\) (rational and not between \(\sqrt{6}\) and \(\sqrt{8}\))
- \(\sqrt{7} \approx 2.64575...\) (irrational)

4. **\(2.7, \sqrt{9}\)**:
- \(2.7\) (rational)
- \(\sqrt{9} = 3\) (rational and not between \(\sqrt{6}\) and \(\sqrt{8}\))

Based on this analysis, the correct option that includes both a rational number and an irrational number between \(\sqrt{6}\) and \(\sqrt{8}\) is:

**\(2.6\bar{6}, \sqrt{7}\)**.