To find a rational number with a denominator of 7 that lies between \(\sqrt{7}\) and \(\sqrt{8}\), we first calculate the approximate values of \(\sqrt{7}\) and \(\sqrt{8}\):
- \(\sqrt{7} \approx 2.645751311\)
- \(\sqrt{8} \approx 2.828427124\)
Next, we look for a rational number in the form of \(\frac{n}{7}\) where \(n\) is an integer that lies between these two values.
To find suitable \(n\), we need to solve the inequalities:
\[ 2.645751311 < \frac{n}{7} < 2.828427124 \]
Multiplying the entire inequality by 7 to eliminate the denominator results in:
\[ 7 \cdot 2.645751311 < n < 7 \cdot 2.828427124 \]
Calculating the products gives:
\[ 18.520258177 < n < 19.798989868 \]
The only integer \(n\) that lies within this range is \(n = 19\).
Thus, the rational number we seek is:
\[ \frac{19}{7} \]
This fraction is already in its simplest form and is between \(\sqrt{7}\) and \(\sqrt{8}\).
So the final answer is:
\[ \frac{19}{7} \]
1 answer