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, π

2.7
,9–√
2 point 7, square root of 9

94
, 7–√
9 fourths, square root of 7

2.6¯¯¯
, 7–√

Answer 1

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

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

So, we need a rational number and an irrational number that lie between approximately 2.45 and 2.83.

Now, let's analyze the given options:

1. **2.5, π**:
- 2.5 is rational, but π (approximately 3.14) is not between \(\sqrt{6}\) and \(\sqrt{8}\).
- This option is not valid.

2. **2.7, \(\sqrt{9}\)**:
- 2.7 is rational, but \(\sqrt{9} = 3\) is also not between \(\sqrt{6}\) and \(\sqrt{8}\).
- This option is not valid.

3. **94, \(\sqrt{7}\)**:
- 94 is an integer (rational), but \(\sqrt{7} \approx 2.64575131106\) is indeed an irrational number and is between \(\sqrt{6}\) and \(\sqrt{8}\).
- However, 94 itself is not between \(\sqrt{6}\) and \(\sqrt{8}\), so this option is not valid.

4. **2.6̅, \(\sqrt{7}\)**:
- 2.6̅, or \(2.\overline{6}\) (which equals \(2.666...\)), is rational and lies between \(\sqrt{6}\) and \(\sqrt{8}\).
- \(\sqrt{7} \approx 2.64575131106\) is irrational and also lies between \(\sqrt{6}\) and \(\sqrt{8}\).

Therefore, the valid option that includes a rational number and an irrational number between \(\sqrt{6}\) and \(\sqrt{8}\) is:

**2.6̅, \(\sqrt{7}\)**.