To determine the bounds for \( \sqrt{97} \) on a number line and find the nearest half, we first approximate \( \sqrt{97} \).
We know:
- \( 9^2 = 81 \)
- \( 10^2 = 100 \)
Hence, \( 9 < \sqrt{97} < 10 \).
Now, we'll narrow it down further. Calculating \( 9.8^2 \): \[ 9.8^2 = 96.04 \] Calculating \( 9.9^2 \): \[ 9.9^2 = 98.01 \]
Thus, we have: \[ 9.8 < \sqrt{97} < 9.9 \]
To find \( \sqrt{97} \) to the nearest half:
- The nearest halves are \( 9.5 \) and \( 10 \).
- Since \( \sqrt{97} \approx 9.848 \), it is closer to \( 10 \).
So, rounding gives: \[ \sqrt{97} \text{ is approximately between } 9.8 \text{ and } 9.9. \]
Thus, in terms of whole numbers and rational numbers to the nearest half:
Whole number: \( 9 \)
Rational number to the nearest half: \( 9.5 \)
So, \( \sqrt{97} \) is between \( 9 \) and \( 9.5 \).
Final answers are: 97−−√ is between 9 and 9.5.