To find a rational number with a denominator of 7 that is between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to approximate the values of \( \sqrt{7} \) and \( \sqrt{8} \).
- \( \sqrt{7} \) is approximately \( 2.64575 \)
- \( \sqrt{8} \) is approximately \( 2.82843 \)
We want a rational number in the form of \( \frac{n}{7} \) such that:
\[ 2.64575 < \frac{n}{7} < 2.82843 \]
Multiplying the entire inequality by 7 to eliminate the denominator gives:
\[ 2.64575 \times 7 < n < 2.82843 \times 7 \]
Calculating the bounds:
- \( 2.64575 \times 7 \approx 18.52025 \)
- \( 2.82843 \times 7 \approx 19.79901 \)
Now we need integer values for \( n \):
The integers between \( 18.52025 \) and \( 19.79901 \) are \( 19 \).
Thus, \( n = 19 \) is a suitable choice. Therefore, the rational number is:
\[ \frac{19}{7} \]
This fraction \( \frac{19}{7} \) is indeed between \( \sqrt{7} \) and \( \sqrt{8} \). In conclusion, the rational number with a denominator of 7 that is between \( \sqrt{7} \) and \( \sqrt{8} \) is:
\[ \boxed{\frac{19}{7}} \]
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