Question

Which of these is a rational number that comes between 7–√ and 8–√ ?

Answer 1

To find a rational number that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to calculate the approximate values of \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).

1. **Calculate \( \sqrt{7} \)**:
\[
\sqrt{7} \approx 2.64575
\]
Thus,
\[
7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425
\]

2. **Calculate \( \sqrt{8} \)**:
\[
\sqrt{8} \approx 2.82843
\]
Thus,
\[
8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157
\]

Now we have:
- \( 7 - \sqrt{7} \approx 4.35425 \)
- \( 8 - \sqrt{8} \approx 5.17157 \)

Next, we need to find a rational number between these two values.

A simple rational number within this range is \( 5 \), as:
\[
4.35425 < 5 < 5.17157
\]

Therefore, \( 5 \) is a rational number that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).