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:
-
56: This is greater than 2.828 and is not between \( \sqrt{7} \) and \( \sqrt{8} \).
-
5 sixths: \( \frac{5}{6} \approx 0.8333 \) which is less than \( \sqrt{7} \) and not between them.
-
197: This is greater than 2.828 and does not fit between \( \sqrt{7} \) and \( \sqrt{8} \).
-
19 sevenths: \( \frac{19}{7} \approx 2.714285714 \) which falls between \( \sqrt{7} \) and \( \sqrt{8} \).
-
34: This number is too large and not in the range.
-
3 fourths: \( \frac{3}{4} = 0.75 \), which is less than \( \sqrt{7} \).
-
63: This number is much greater than \( \sqrt{8} \).
-
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.