To find a rational number with a denominator of 7 that lies between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to approximate the values of \( \sqrt{7} \) and \( \sqrt{8} \).
Calculating the square roots:
\[ \sqrt{7} \approx 2.6457513110645906 \] \[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \approx 2.8284271247461903 \]
Now, we need to find a rational number of the form \( \frac{n}{7} \) (where \( n \) is an integer) that lies between these two values.
To meet the conditions, we set up the following inequality:
\[ \sqrt{7} < \frac{n}{7} < \sqrt{8} \]
Multiplying the entire inequality by 7 gives us:
\[ 7\sqrt{7} < n < 7\sqrt{8} \]
Calculating the bounds:
\[ 7\sqrt{7} \approx 7 \times 2.6457513110645906 \approx 18.520258177452134 \] \[ 7\sqrt{8} = 7 \times 2.8284271247461903 \approx 19.79898987624445 \]
Now, since \( n \) must be an integer, we can look at the integers between 18.520 and 19.798. The only integer that fits is \( n = 19 \).
Thus, our rational number is:
\[ \frac{19}{7} \]
Finally, we can confirm that \( \frac{19}{7} \) lies between \( \sqrt{7} \) and \( \sqrt{8} \):
- \( \sqrt{7} \approx 2.6457513110645906 \)
- \( \frac{19}{7} \approx 2.7142857142857144 \)
- \( \sqrt{8} \approx 2.8284271247461903 \)
Since:
\[ \sqrt{7} < \frac{19}{7} < \sqrt{8} \]
Therefore, 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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