To find a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{7} \), we first need to calculate \( 7 - \sqrt{7} \) and \( 8 - \sqrt{7} \).
-
Calculate approximations:
- The square root of 7 is approximately \( \sqrt{7} \approx 2.6457513110645906 \).
- Thus: \[ 7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.354248688935409 \] \[ 8 - \sqrt{7} \approx 8 - 2.6457513110645906 \approx 5.354248688935409 \]
-
Set up the inequality: We want a rational number \( \frac{a}{7} \) such that: \[ 4.354248688935409 < \frac{a}{7} < 5.354248688935409 \]
-
Multiply through by 7: Multiplying the entire inequality by 7 results in: \[ 7 \times 4.354248688935409 < a < 7 \times 5.354248688935409 \] Which simplifies to: \[ 30.480740822548863 < a < 37.48074082254886 \]
-
Identify integer values for \( a \): The integer values for \( a \) must be between 31 and 37:
- Possible integer values for \( a \) are \( 31, 32, 33, 34, 35, 36, 37 \).
-
Select a value for \( a \): Any of these values will yield a rational number with the desired properties, but we can select \( a = 34 \) for simplicity.
Thus, the rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{7} \) is:
\[ \frac{34}{7} \]