To find a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to approximate the two values.
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Compute \( 7 - \sqrt{7} \):
- 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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Compute \( 8 - \sqrt{8} \):
- The approximate value of \( \sqrt{8} \) is about 2.82843.
- Therefore, \( 8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157 \).
Now we know that we need a rational number between approximately 4.35425 and 5.17157 with a denominator of 7.
- Convert possible rational numbers:
- To find a suitable fraction, we check numbers of the form \( \frac{n}{7} \), where \( n \) is an integer.
- We can multiply our bounds by 7 to convert these to integer values:
- \( 4.35425 \times 7 \approx 30.5 \)
- \( 5.17157 \times 7 \approx 36.2 \)
Thus, we need \( n \) to satisfy: \[ 30.5 < n < 36.2 \]
The integers \( n \) that fall within this range are \( 31, 32, 33, 34, 35, \) and \( 36 \).
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Choosing a rational number:
- Let's pick \( n = 35 \): \[ \frac{35}{7} = 5 \]
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Check if it lies strictly between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \):
- Since \( 5 \) is between \( 4.35425 \) and \( 5.17157 \), it satisfies our condition.
Thus, a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \) is:
\[ \frac{35}{7} \]