To find a rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\), we first need to find the approximate values of \(\sqrt{7}\) and \(\sqrt{8}\).
Calculating these values:
- \(\sqrt{7} \approx 2.64575\)
- \(\sqrt{8} = 2\sqrt{2} \approx 2.82843\)
Next, we need to express a rational number \( \frac{n}{7} \) that falls between these two values.
To find \( n \), we can set up the following inequalities: \[ 2.64575 < \frac{n}{7} < 2.82843 \]
Multiplying the entire inequality by 7 gives: \[ 7 \times 2.64575 < n < 7 \times 2.82843 \] This simplifies to: \[ 18.52025 < n < 19.79801 \]
Since \( n \) must be an integer, the only integer that fits this condition is \( n = 19 \).
Thus, the rational number is: \[ \frac{19}{7} \]
To verify:
- Calculate \( \frac{19}{7} \) which is approximately \( 2.7142857 \). This value is indeed between \( \sqrt{7} \) (approximately \( 2.64575 \)) and \( \sqrt{8} \) (approximately \( 2.82843 \)).
Therefore, the rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\) is: \[ \frac{19}{7} \]
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