To determine the whole number and the rational number (to the nearest half) that \(\sqrt{97}\) falls between, we can start by estimating the value of \(\sqrt{97}\).
- Identify the perfect squares around 97:
- \(9^2 = 81\)
- \(10^2 = 100\)
From this, we know that: \[ 9 < \sqrt{97} < 10 \]
- Finding a more precise estimate:
- We can start calculating some squares to find a value closer to 97.
- \(9.5^2 = 90.25\)
- \(9.7^2 = 94.09\)
- \(9.8^2 = 96.04\)
- \(9.9^2 = 98.01\)
Since \(9.8^2\) is just below 97 and \(9.9^2\) is just above 97, we can say: \[ 9.8 < \sqrt{97} < 9.9 \]
- Rational number to the nearest half:
- The nearest halves around \(9.8\) would be \(9.5\) and \(10.0\).
- Since \(\sqrt{97} \approx 9.848\), it rounds to \(9.5\).
In conclusion, \(\sqrt{97}\) is between the whole number \(9\) and the rational number \(9.5\) (to the nearest half).
So, your answer is that \(\sqrt{97}\) is between 9 and 9.5.