To find a rational number with a denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \), we first calculate the values of these square roots:
\[ \sqrt{8} \approx 2.828 \] \[ \sqrt{9} = 3 \]
Now we are looking for a rational number in the form \( \frac{n}{7} \) such that:
\[ 2.828 < \frac{n}{7} < 3 \]
To find the corresponding values for \( n \), we can multiply the inequalities by 7:
\[ 7 \times 2.828 < n < 7 \times 3 \]
Calculating these products:
\[ 19.796 < n < 21 \]
Since \( n \) must be an integer, the possible integer value for \( n \) is 20.
Now, we can check this value:
\[ \frac{20}{7} \approx 2.857 \]
Since \( 2.828 < 2.857 < 3 \), we find that \( \frac{20}{7} \) is indeed between \( \sqrt{8} \) and \( \sqrt{9} \).
Thus, the rational number with the denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \) is:
\[ \frac{20}{7} \]