To find a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to evaluate the decimal approximations of \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).
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Calculate \( 7 - \sqrt{7} \):
- The approximate value of \( \sqrt{7} \) is about \( 2.6457513110645906 \).
- So, \( 7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.3542486889354094 \).
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Calculate \( 8 - \sqrt{8} \):
- The approximate value of \( \sqrt{8} \) is about \( 2.8284271247461903 \).
- So, \( 8 - \sqrt{8} \approx 8 - 2.8284271247461903 \approx 5.1715728752538097 \).
Now we need a rational number in the form of an improper fraction with a denominator of 7 that lies between approximately \( 4.3542486889354094 \) and \( 5.1715728752538097 \).
To make this simpler, we can express the numbers between 4 and 5 in terms of fractions with a denominator of 7. This means we are looking for fractions like:
\[ \frac{n}{7} \]
Where \( n \) is an integer.
The integers that correspond to rational numbers between these bounds multiplied by 7:
- \( 4.354 \times 7 \approx 30.478 \)
- \( 5.171 \times 7 \approx 36.197 \)
We can round these bounds to their nearest integers:
- Lower bound: \( 30 \)
- Upper bound: \( 36 \)
The integers between 30 and 36 are \( 31, 32, 33, 34, 35, 36 \). Now we convert these into improper fractions with a denominator of 7:
- \( \frac{31}{7} \approx 4.42857 \) (valid)
- \( \frac{32}{7} \approx 4.57143 \) (valid)
- \( \frac{33}{7} \approx 4.71429 \) (valid)
- \( \frac{34}{7} \approx 4.85714 \) (valid)
- \( \frac{35}{7} \approx 5.00000 \) (valid)
- \( \frac{36}{7} \approx 5.14286 \) (valid)
Therefore, any of these fractions—\( \frac{31}{7}, \frac{32}{7}, \frac{33}{7}, \frac{34}{7}, \frac{35}{7}, \frac{36}{7} \)—are valid answers.
For instance, if you take \( \frac{33}{7} \):
\[ \frac{33}{7} \approx 4.71429 \]
This rational number is between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).
Thus, one possible answer is:
\[ \boxed{\frac{33}{7}} \]