To find a rational number with a denominator of 7 that is between \( \sqrt{7} \) and \( \sqrt{8} \), we start by determining the decimal values of \( \sqrt{7} \) and \( \sqrt{8} \).
First, we evaluate these square roots:
- \( \sqrt{7} \approx 2.64575 \)
- \( \sqrt{8} = 2\sqrt{2} \approx 2.82843 \)
Now we need to find a rational number between these two values. A rational number with a denominator of 7 can be expressed in the form \( \frac{n}{7} \) for some integer \( n \).
Next, we will find an integer \( n \) such that: \[ 2.64575 < \frac{n}{7} < 2.82843. \] To find appropriate bounds for \( n \), we multiply the inequalities by 7:
For the lower bound: \[ 2.64575 \times 7 \approx 18.52025, \] which implies \( n > 18.52025 \). Therefore, \( n \) must be at least \( 19 \).
For the upper bound: \[ 2.82843 \times 7 \approx 19.79901, \] which implies \( n < 19.79901 \). Therefore, \( n \) can be at most \( 19 \).
The only integer value for \( n \) that fits both inequalities is \( n = 19 \).
Thus, the rational number in the required form is: \[ \frac{19}{7}. \]
This is the rational number with a denominator of 7 that lies between \( \sqrt{7} \) and \( \sqrt{8} \).
Therefore, the final answer is: \[ \frac{19}{7}. \]
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