Question

Which of the following options includes a rational number and an irrational number between √(6) and √(8)?
A. (9)/(4), √(7)
B. 2.7, √(9)
C. \overline{6}, √(7)
D. 2.5, \pi

Answer 1

wrong

Answer 2

To determine which option includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \):

1. **Calculate \( \sqrt{6} \) and \( \sqrt{8} \)**:
- \( \sqrt{6} \) is approximately \( 2.45 \).
- \( \sqrt{8} \) is approximately \( 2.83 \).

Therefore, we are looking for one rational number and one irrational number that lie within the interval \( (2.45, 2.83) \).

2. **Evaluate the options**:

- **Option A**: \( \frac{9}{4} \) and \( \sqrt{7} \)
- \( \frac{9}{4} = 2.25 \) (rational, outside the interval)
- \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

- **Option B**: \( 2.7 \) and \( \sqrt{9} \)
- \( 2.7 \) (rational, inside the interval)
- \( \sqrt{9} = 3 \) (rational, outside the interval)

- **Option C**: \( \overline{6} = 6.\overline{6} \) (6.666...) and \( \sqrt{7} \)
- \( 6.\overline{6} \) (rational, outside the interval)
- \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

- **Option D**: \( 2.5 \) and \( \pi \)
- \( 2.5 \) (rational, inside the interval)
- \( \pi \approx 3.14159 \) (irrational, outside the interval)

3. **Conclusion**:
Only **Option A** \( \frac{9}{4} \, and \, \sqrt{7} \) contains one rational number (\(\sqrt{7}\)) and one irrational number. However, \(\frac{9}{4}\) is outside the specified range. All other options mix rational and irrational numbers but also fall outside the range.

**Thus, the best choice is A, where only one number is irrational within the range.**

Answer 3

Let's reevaluate the options carefully.

We want to find a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \). As established:

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

**Now, we will assess each option again:**

### Option A: \( \frac{9}{4}, \sqrt{7} \)
- \( \frac{9}{4} = 2.25 \) (rational, outside the interval)
- \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

### Option B: \( 2.7, \sqrt{9} \)
- \( 2.7 \) (rational, inside the interval)
- \( \sqrt{9} = 3 \) (rational, outside the interval)

### Option C: \( 6.\overline{6}, \sqrt{7} \)
- \( 6.\overline{6} \) (rational, outside the interval)
- \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

### Option D: \( 2.5, \pi \)
- \( 2.5 \) (rational, inside the interval)
- \( \pi \approx 3.14159 \) (irrational, outside the interval)

### Summary of Findings:
- **Option A:** One number is irrational, but the rational number is not in the interval.
- **Option B:** One number is in the interval, but both numbers cannot meet the criteria since \( \sqrt{9} \) is rational and outside the interval.
- **Option C:** The rational number is outside the interval, while the irrational number is inside.
- **Option D:** Here, \( 2.5 \) is in the interval and \( \pi \) is outside.

### Correct Conclusion:
The only option where the irrational number is indeed in the interval \( ( \sqrt{6}, \sqrt{8} ) \) is **Option C** which contains \( \sqrt{7} \) as the irrational number.

However, it seems there was an oversight in keeping track of which option provided both a rational and an irrational number correctly fitting the interval criteria since:

- Options with **both numbers fit** the criteria of one inside:

- We can state that **Option A** and **Option B** do not have a corresponding rational number that stays within \( ( \sqrt{6}, \sqrt{8} )**.

So, the best correct pairs that do qualify for being between \(\sqrt{6}\) and \(\sqrt{8}\) are actually **Option D: \(2.5, \pi\)** as rational and irrational, respectively, given that both recognize \(2.5\) inside the range and \( \pi \) outside.

Thank you for your patience while correcting this!