To find a rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\), we first need to calculate the approximate decimal values of \(\sqrt{7}\) and \(\sqrt{8}\).
- Calculate \(\sqrt{7}\) and \(\sqrt{8}\):
- \(\sqrt{7} \approx 2.64575\)
- \(\sqrt{8} \approx 2.82843\)
Now, we need to find a rational number in the form \(\frac{n}{7}\) that falls between these two values. This can be done by setting up the inequalities:
\[ \sqrt{7} < \frac{n}{7} < \sqrt{8} \]
Multiplying all sides of the inequalities by 7 gives us:
\[ 7\sqrt{7} < n < 7\sqrt{8} \]
Calculating \(7\sqrt{7}\) and \(7\sqrt{8}\):
- \(7\sqrt{7} \approx 7 \times 2.64575 \approx 18.52025\)
- \(7\sqrt{8} \approx 7 \times 2.82843 \approx 19.79901\)
This means we need to find integers \(n\) such that:
\[ 18.52025 < n < 19.79901 \]
The only integer \(n\) that satisfies this inequality is \(n=19\).
Thus, the rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\) is:
\[ \frac{19}{7} \]
This is the improper fraction you were looking for.