To find a rational number with a denominator of 7 that is between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to calculate the approximate values of these expressions.
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Calculate \( 7 - \sqrt{7} \):
- The approximate value of \( \sqrt{7} \) is about \( 2.64575 \).
- Thus, \( 7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425 \).
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Calculate \( 8 - \sqrt{8} \):
- The approximate value of \( \sqrt{8} \) is \( 2\sqrt{2} \approx 2.82843 \).
- Therefore, \( 8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157 \).
Now, we are looking for a rational number with a denominator of 7 that lies between approximately \( 4.35425 \) and \( 5.17157 \).
Rational numbers with a denominator of 7 can be expressed in the form \( \frac{n}{7} \), where \( n \) is an integer.
To find suitable integers \( n \):
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Multiply the lower bound \( 4.35425 \) by 7: \[ 4.35425 \times 7 \approx 30.48075. \] This suggests that \( n \) must be greater than 30.
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Multiply the upper bound \( 5.17157 \) by 7: \[ 5.17157 \times 7 \approx 36.201. \] This suggests that \( n \) must be less than 36.
Thus, \( n \) can take on the values \( 31, 32, 33, 34, 35, \) or \( 36 \).
Choosing any of these values gives us the rational numbers:
- \( \frac{31}{7} \approx 4.42857 \)
- \( \frac{32}{7} \approx 4.57143 \)
- \( \frac{33}{7} \approx 4.71429 \)
- \( \frac{34}{7} \approx 4.85714 \)
- \( \frac{35}{7} \approx 5.00000 \)
- \( \frac{36}{7} \approx 5.14286 \)
All of these values fall between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).
So, we can take any of these fractions as our final answer. A suitable answer could be \( \frac{34}{7} \).
Therefore, the answer is: \[ \frac{34}{7} \]