To find a rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\), we first need to determine the approximate values of \(\sqrt{7}\) and \(\sqrt{8}\).
Calculating the square roots:
- \(\sqrt{7} \approx 2.6457513110645906\)
- \(\sqrt{8} \approx 2.8284271247461903\)
Now, we want to find a rational number in the form of \(\frac{n}{7}\) (where \(n\) is an integer) that falls within the interval \((\sqrt{7}, \sqrt{8})\).
Setting the inequalities:
\[ \sqrt{7} < \frac{n}{7} < \sqrt{8} \]
Multiplying all parts of the inequality by 7 (since 7 is positive and doesn't change the direction of the inequality):
\[ 7\sqrt{7} < n < 7\sqrt{8} \]
Calculating the bounds:
- \(7\sqrt{7} \approx 7 \times 2.6457513110645906 \approx 18.520258177451134\)
- \(7\sqrt{8} \approx 7 \times 2.8284271247461903 \approx 19.79898987322333\)
Now, we need integers \(n\) that fall within the range \(18.520258177451134 < n < 19.79898987322333\). The only integer \(n\) that satisfies this condition is \(n = 19\).
Thus, the rational number with a denominator of 7 is:
\[ \frac{19}{7} \]
Finally, we can confirm that:
\[ \sqrt{7} \approx 2.6457513110645906 < \frac{19}{7} \approx 2.7142857142857144 < \sqrt{8} \approx 2.8284271247461903 \]
Therefore, the answer is:
\[ \frac{19}{7} \]
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