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To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we can convert the given square roots into rational numbers with denominators of 7.

7–√ can be written as 7/√7 * √7 = 7√7 / 7 = √7.

8–√ can be written as 8/√7 * √7 = 8√7 / 7.

To find a rational number between √7 and 8√7 / 7, we can take their average:

(√7 + 8√7 / 7) / 2 = (2√7 + 8√7) / 14 = 10√7 / 14 = 5√7 / 7.

Therefore, a rational number with a denominator of 7 that is between 7–√ and 8–√ is 5√7 / 7, written as an improper fraction.

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Answer:

Yes, the given solution is correct.

Explanation:

To find a rational number with a denominator of 7 between 7–√ and 8–√, we can convert the given square roots into rational numbers with denominators of 7.

First, we convert 7–√ into a rational number with a denominator of 7. We do this by multiplying it by √7/√7, which is equal to 1. So, 7–√ can be written as (7/√7) * (√7/√7) = 7√7 / 7 = √7.

Next, we convert 8–√ into a rational number with a denominator of 7. Similarly, we multiply it by √7/√7, which is equal to 1. So, 8–√ can be written as (8/√7) * (√7/√7) = 8√7 / 7.

Now, we have the rational numbers √7 and 8√7 / 7. To find a rational number between them, we can take their average. We add the two numbers and divide by 2:

(√7 + 8√7 / 7) / 2 = (2√7 + 8√7) / 14 = 10√7 / 14 = 5√7 / 7.

Therefore, a rational number with a denominator of 7 that is between 7–√ and 8–√ is 5√7 / 7, written as an improper fraction.

This solution follows the correct steps to find a rational number between the given square roots and explains each step clearly.

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