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

and 11−−√
.

2 answers

To find a rational number that is between \( \sqrt{10} \) and \( \sqrt{11} \), we first need to approximate the values of these square roots:

  • \( \sqrt{10} \) is approximately \( 3.162 \) (more precisely, it's roughly \( 3.162277660168379 \)).
  • \( \sqrt{11} \) is approximately \( 3.317 \) (more precisely, it's roughly \( 3.3166247903554 \)).

Next, we look for a rational number between these two values. A simple choice would be the rational number \( \frac{32}{10} \) or \( 3.2 \).

To verify:

  • \( 3.162 < 3.2 < 3.317 \)

Thus, \( \frac{32}{10} \) or simply \( 3.2 \) is a rational number between \( \sqrt{10} \) and \( \sqrt{11} \). Other rational numbers could include \( \frac{31}{10} = 3.1 \) or \( \frac{33}{10} = 3.3 \) (3.3 is not acceptable since it exceeds \( \sqrt{11} \)).

In summary, \( \frac{32}{10} \) or simply \( 3.2 \) is a rational number that satisfies the requirement.

only answers are 16/5 10/3 3.1 3.4 which is correct?