To find a rational number between $7-\sqrt7$ and $8-\sqrt7$, we can average the two numbers. Averaging is equivalent to finding the midpoint on a number line. The number $\frac{7-\sqrt7 + 8-\sqrt7}{2}$ is the average or midpoint of $7-\sqrt7$ and $8-\sqrt7$. We simplify the numerator and write it as a decimal to calculate the average:
\begin{align*}
\frac{7-\sqrt7 + 8-\sqrt7}{2} &= \frac{7+8-2\sqrt7}{2} \\
&= \frac{15-2\sqrt7}{2} \\
&= 7.5-\sqrt7.
\end{align*}Notice that the resulting number is $7.5-\sqrt7$. To express this number as a fraction with a denominator of 7, we can multiply the numerator and denominator by 7 to get \[
\frac{(7.5-\sqrt7)\cdot7}{1\cdot7}=\boxed{52-7\sqrt7}.
\]
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.
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