To find a rational number between \( \sqrt{10} \) and \( \sqrt{11} \), we first need to estimate the values of these square roots.
Calculating \( \sqrt{10} \): \[ \sqrt{10} \approx 3.162 \]
Calculating \( \sqrt{11} \): \[ \sqrt{11} \approx 3.317 \]
Now that we have approximate values for \( \sqrt{10} \) and \( \sqrt{11} \), we can find a rational number between these two values.
One simple rational number we can choose is \( 3.2 \).
Now we can verify that \( 3.2 \) is indeed between \( \sqrt{10} \) and \( \sqrt{11} \):
- \( 3.162 < 3.2 < 3.317 \)
Thus, \( 3.2 \) is a rational number that lies between \( \sqrt{10} \) and \( \sqrt{11} \). Another example could be \( \frac{32}{10} \) or \( \frac{16}{5} \), which is equivalent to \( 3.2 \).
Therefore, you can use \( 3.2 \) or \( \frac{16}{5} \) as a rational number between \( \sqrt{10} \) and \( \sqrt{11} \).
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