To find a rational number with a denominator of 7 that is between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to calculate \( \sqrt{7} \) and \( \sqrt{8} \).
Calculating these values:
\[ \sqrt{7} \approx 2.6457513110645906 \] \[ \sqrt{8} = 2\sqrt{2} \approx 2.8284271247461903 \]
Next, we note that we want to find a rational number of the form \( \frac{a}{7} \) for some integer \( a \) that falls between \( \sqrt{7} \) and \( \sqrt{8} \). This means we need to solve the inequalities:
\[ \sqrt{7} < \frac{a}{7} < \sqrt{8} \]
Multiplying all parts of this inequality by 7 gives:
\[ 7\sqrt{7} < a < 7\sqrt{8} \]
Calculating the lower and upper bounds:
\[ 7\sqrt{7} \approx 7 \times 2.6457513110645906 \approx 18.520258177452134 \] \[ 7\sqrt{8} = 7 \times 2.8284271247461903 \approx 19.79898987322333 \]
Thus, we have:
\[ 18.520258177452134 < a < 19.79898987322333 \]
Since \( a \) must be an integer, the only integer \( a \) that satisfies this inequality is \( 19 \).
Therefore, the rational number we seek is:
\[ \frac{19}{7} \]
To verify, we check if \( \frac{19}{7} \) lies between \( \sqrt{7} \) and \( \sqrt{8} \):
Calculating \( \frac{19}{7} \):
\[ \frac{19}{7} \approx 2.7142857142857144 \]
Now we check the inequalities:
\[ \sqrt{7} \approx 2.6457513110645906 < 2.7142857142857144 < 2.8284271247461903 \approx \sqrt{8} \]
Thus, \( \frac{19}{7} \) is indeed between \( \sqrt{7} \) and \( \sqrt{8} \).
The answer is:
\[ \boxed{\frac{19}{7}} \]
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