Show an example of a rational number that is between 10−−√ and 11−−√

Answer 1

To find a rational number between \( \sqrt{10} \) and \( \sqrt{11} \), we first need to evaluate these square roots.

1. Calculate \( \sqrt{10} \) and \( \sqrt{11} \):

   • \( \sqrt{10} \approx 3.162 \)
   • \( \sqrt{11} \approx 3.317 \)

2. Identify rational numbers in this range:

   Any rational number between these two values should be greater than approximately \( 3.162 \) and less than approximately \( 3.317 \).

3. Choose a rational number:

   A simple rational number that falls between these values is \( \frac{32}{10} \) or \( 3.2 \).

4. Verification:

   • \( 3.2 \) is indeed greater than \( \sqrt{10} \approx 3.162 \)
   • \( 3.2 \) is also less than \( \sqrt{11} \approx 3.317 \)

Thus, an example of a rational number between \( \sqrt{10} \) and \( \sqrt{11} \) is \( \frac{32}{10} \) or simply \( 3.2 \).