Question

Which of these is a rational number that comes between 7–√ and 8–√ ?(1 point)

Answer 1

To find a rational number between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to evaluate these square roots to approximate their values.

- The value of \( \sqrt{7} \) is approximately 2.6457513110645906.
- The value of \( \sqrt{8} \) is \( 2\sqrt{2} \), which is approximately 2.8284271247461903.

To find a rational number between these two values, we can take the average of \( \sqrt{7} \) and \( \sqrt{8} \):

\[
\frac{\sqrt{7} + \sqrt{8}}{2}
\]

Calculating this:

\[
\frac{2.6457513110645906 + 2.8284271247461903}{2} \approx \frac{5.474178435810781}{2} \approx 2.7370892179053905
\]

This value \( 2.7370892179053905 \) is between \( \sqrt{7} \) and \( \sqrt{8} \), but we need a rational number.

A simple rational number that falls between \( \sqrt{7} \) and \( \sqrt{8} \) is \( \frac{11}{4} = 2.75 \), since:

- \( \sqrt{7} \approx 2.64575 < \frac{11}{4} = 2.75 < \sqrt{8} \approx 2.82843 \)

Thus, \( \frac{11}{4} \) is a rational number that lies between \( \sqrt{7} \) and \( \sqrt{8} \).