To find a rational number with a denominator of 7 that lies between \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\), we'll first calculate the approximate values of \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\).
- Calculate \(\sqrt{7} \approx 2.64575\).
- Therefore, \[ 7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425 \] and \[ 8 - \sqrt{7} \approx 8 - 2.64575 \approx 5.35425. \]
Now we know we are looking for a rational number that is between approximately \(4.35425\) and \(5.35425\).
A rational number in the form of \(\frac{n}{7}\) that lies in this range means we want: \[ 4.35425 < \frac{n}{7} < 5.35425. \]
Next, we can multiply all parts of the inequality by 7 (since 7 is a positive number, it does not change the inequality direction): \[ 7 \times 4.35425 < n < 7 \times 5.35425. \] Calculating those bounds: \[ 30.48 < n < 37.48. \]
Thus, \(n\) must be an integer between 31 and 37. The possible integer values of \(n\) are 31, 32, 33, 34, 35, 36, and 37.
Let’s select \(n = 34\): \[ \frac{34}{7}. \] We now check if this value is between \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\).
Calculating \(\frac{34}{7}\): \[ \frac{34}{7} \approx 4.85714. \]
Since \(4.35425 < 4.85714 < 5.35425\), we confirm that \(\frac{34}{7}\) lies within that range.
Thus, the answer is: \[ \frac{34}{7}. \]