To find a rational number with a denominator of 7 that lies between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to calculate the approximate values of \( \sqrt{7} \) and \( \sqrt{8} \):
- \( \sqrt{7} \approx 2.645751311 \)
- \( \sqrt{8} \approx 2.828427124 \)
Now, we want to find a rational number in the form of an improper fraction with a denominator of 7 that falls between these two values.
We can express a rational number as \( \frac{n}{7} \), where \( n \) is an integer. We can derive the bounds for n by multiplying by 7:
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For \( \sqrt{7} \): \[ n > 7 \cdot \sqrt{7} \approx 7 \cdot 2.645751311 \approx 18.520258177 \]
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For \( \sqrt{8} \): \[ n < 7 \cdot \sqrt{8} \approx 7 \cdot 2.828427124 \approx 19.799, \text{ rounding down gives } n < 19 \]
Thus, \( n \) can take values of 19 and above. The only integer value satisfying \( 18.52 < n < 19.79 \) is 19.
Now we can express this as a rational number: \[ \frac{19}{7} \]
So, the improper fraction that lies between \( \sqrt{7} \) and \( \sqrt{8} \) is: \[ \frac{19}{7} \]
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