To determine between which two rational numbers (to the nearest half) the square root of 97 falls, we first calculate the approximate value of \(\sqrt{97}\).
We know:
- \(\sqrt{81} = 9\)
- \(\sqrt{100} = 10\)
Since \(97\) is between \(81\) and \(100\), it follows that: \[ 9 < \sqrt{97} < 10 \]
Next, we can refine this estimate. We can check the squares of numbers between \(9\) and \(10\):
Calculate \(9.5^2\): \[ 9.5^2 = 90.25 \] Calculate \(9.6^2\): \[ 9.6^2 = 92.16 \] Calculate \(9.7^2\): \[ 9.7^2 = 94.09 \] Calculate \(9.8^2\): \[ 9.8^2 = 96.04 \] Calculate \(9.9^2\): \[ 9.9^2 = 98.01 \]
Since \(9.8^2 = 96.04\) and \(9.9^2 = 98.01\), it shows: \[ 9.8 < \sqrt{97} < 9.9 \]
To express \(9.8\) and \(9.9\) to the nearest half, we observe that \(9.75\) would also fit between them:
- Therefore, we can conclude: \[ \sqrt{97} \text{ is between } 9.5 \text{ and } 10. \]
Thus, the final answer is that \(\sqrt{97}\) is between \(9.5\) and \(10\).