To find a rational number with a denominator of 7 that lies between \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\), we first calculate the approximate values of these expressions.
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Approximate \( \sqrt{7} \): \[ \sqrt{7} \approx 2.64575 \]
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Calculate \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\)**: \[ 7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425 \] \[ 8 - \sqrt{7} \approx 8 - 2.64575 \approx 5.35425 \]
Now we need a rational number that is between approximately \(4.35425\) and \(5.35425\) with a denominator of \(7\).
We can express a number with a denominator of 7 in the form of \( \frac{n}{7} \) where \( n \) is an integer. We want to find integers \( n \) such that: \[ 4.35425 < \frac{n}{7} < 5.35425 \]
To find the corresponding integers for \(n\), we multiply the inequalities by 7: \[ 4.35425 \times 7 < n < 5.35425 \times 7 \] Calculating these: \[ 4.35425 \times 7 \approx 30.48775 \] \[ 5.35425 \times 7 \approx 37.48075 \]
Thus, \(n\) must satisfy: \[ 30.48775 < n < 37.48075 \]
The integers that satisfy this inequality are \(31, 32, 33, 34, 35, 36, 37\).
Converting these integers into their corresponding fractions gives us:
- \(31/7\)
- \(32/7\)
- \(33/7\)
- \(34/7\)
- \(35/7\)
- \(36/7\)
- \(37/7\)
Among these, we should identify which ones fall between \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\):
- \(31/7 \approx 4.42857\) (valid)
- \(32/7 \approx 4.57143\) (valid)
- \(33/7 \approx 4.71429\) (valid)
- \(34/7 \approx 4.85714\) (valid)
- \(35/7 \approx 5.0\) (valid)
- \(36/7 \approx 5.14286\) (valid)
- \(37/7 \approx 5.28571\) (valid)
Thus, every value is valid between \(4.35425\) and \(5.35425\). Any of these can be chosen as an answer.
A specific example is: \[ \frac{34}{7} \]
Therefore, one rational number with a denominator of 7 that lies between \(7 - \sqrt{7}\) and \(8 - \sqrt{7}\) is \(\boxed{\frac{34}{7}}\).
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