To find a rational number between 7–√ and 8–√ with a denominator of 7, we can use the fact that (√2)^2 = 2.
Let's subtract (√2)^2 = 2 from both sides of the inequality:
7 - √2 - 2 < x < 8 - √2 - 2
This simplifies to:
5 - √2 < x < 6 - √2
Now, let's rationalize the denominator by multiplying both sides of the inequality by 7:
7(5 - √2) < 7x < 7(6 - √2)
This gives:
35 - 7√2 < 7x < 42 - 7√2
Divide all three sides by 7:
35/7 - 7√2/7 < x < 42/7 - 7√2/7
Simplifying further:
5 - √2 < x < 6 - √2
Therefore, a rational number between 7–√ and 8–√ with a denominator of 7 is in the range 5 - √2 < x < 6 - √2.
However, it is not possible to write this as an improper fraction with a denominator of 7, since the range is in terms of √2 and cannot be expressed as a simple fraction.