Question

Which of these is a rational number that comes between 7–√
and 8–√
?(1 point)
Responses

56
5 sixths

197
19 sevenths

34
3 fourths

63
6 thirds

Answer 1

To determine which of the given options is a rational number between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to approximate the square roots.

- \( \sqrt{7} \approx 2.645751311 \)
- \( \sqrt{8} \approx 2.828427124 \)

Therefore, we are looking for a rational number that is approximately between 2.645 and 2.828.

Now let's analyze the given options:

1. **56:** This is greater than 2.828 and is not between \( \sqrt{7} \) and \( \sqrt{8} \).

2. **5 sixths:** \( \frac{5}{6} \approx 0.8333 \) which is less than \( \sqrt{7} \) and not between them.

3. **197:** This is greater than 2.828 and does not fit between \( \sqrt{7} \) and \( \sqrt{8} \).

4. **19 sevenths:** \( \frac{19}{7} \approx 2.714285714 \) which falls between \( \sqrt{7} \) and \( \sqrt{8} \).

5. **34:** This number is too large and not in the range.

6. **3 fourths:** \( \frac{3}{4} = 0.75 \), which is less than \( \sqrt{7} \).

7. **63:** This number is much greater than \( \sqrt{8} \).

8. **6 thirds:** \( \frac{6}{3} = 2 \), which is also less than \( \sqrt{7} \).

Thus, the only option that fits the criteria for a rational number between \( \sqrt{7} \) and \( \sqrt{8} \) is **19 sevenths**.