Question

Which of the following options includes a rational number and an irrational number between 6–√
and 8–√
?(1 point)
Responses

2.5, π
2.5 comma pi

2.6¯¯¯, 7–√
2 point Modifying above 6 with bar comma start root 7 end root

2.7, 9–√
2.7 comma start root 9 end root

94, 7–√

Answer 1

To determine which option includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \), we first find the approximate values of these square roots:

- \( \sqrt{6} \) is approximately \( 2.45 \)
- \( \sqrt{8} \) is approximately \( 2.83 \)

Now let's analyze each option:

1. **2.5, π**:
- \( 2.5 \) is rational and \( π \) is irrational.
- \( 2.5 \) is between \( \sqrt{6} \) and \( \sqrt{8} \).

2. **2.6, \( \sqrt{7} \)**:
- \( 2.6 \) is rational and \( \sqrt{7} \) is irrational.
- \( 2.6 \) is between \( \sqrt{6} \) and \( \sqrt{8} \) since \( \sqrt{7} \) is also in this range.

3. **2.7, \( \sqrt{9} \)**:
- \( 2.7 \) is rational and \( \sqrt{9} = 3 \) is rational.
- This option does not include an irrational number.

4. **94, \( \sqrt{7} \)**:
- \( 94 \) is rational, but it is not between \( \sqrt{6} \) and \( \sqrt{8} \), and \( \sqrt{7} \) is irrational.

From the analysis above, both options 1 and 2 include one rational and one irrational number that lies between \( \sqrt{6} \) and \( \sqrt{8} \). However, since only one option is typically required in a question like this:

- **2.6, \( \sqrt{7} \)** is the best option as both values are also within the specified range.

So the answer could be either option 1 or option 2, but option 2 specifically meets the criteria strictly between \( \sqrt{6} \) and \( \sqrt{8} \) with both types of numbers.

Therefore, the best answer is:

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