To determine the whole number and rational number to the nearest half that \( \sqrt{97} \) is between, we first need to find the approximate value of \( \sqrt{97} \).
Calculating \( \sqrt{97} \):
- We know that \( 9^2 = 81 \) and \( 10^2 = 100 \). Thus, \( \sqrt{97} \) is between 9 and 10.
- To get a more precise value, we can estimate. Since \( 97 \) is closer to \( 100 \) than to \( 81 \), we estimate that \( \sqrt{97} \) is closer to \( 10 \).
Calculating further:
- By trying \( 9.8 \): \( 9.8^2 = 96.04 \) (too low)
- Trying \( 9.9 \): \( 9.9^2 = 98.01 \) (too high)
That's enough to conclude that: \[ 9.8 < \sqrt{97} < 9.9 \]
To express \( \sqrt{97} \) to the nearest half:
- The nearest halves around 9.8 are 9.5 and 10.
- Since \( \sqrt{97} \) is approximately 9.8, it rounds to 10.
Combining this information, we find:
- The whole number is 9 (the largest whole number less than \( \sqrt{97} \)).
- The nearest rational number to half is 10.
Therefore, \( \sqrt{97} \) is between: 9 and 10.
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