To find between which whole number and rational number (to the nearest half) the square root of 97 (\(\sqrt{97}\)) lies, we first calculate the approximate value of \(\sqrt{97}\).
The whole numbers we can consider are:
- \(\sqrt{81} = 9\) (since \(9^2 = 81\))
- \(\sqrt{100} = 10\) (since \(10^2 = 100\))
Since \(97\) is between \(81\) and \(100\), we can conclude that:
\[ 9 < \sqrt{97} < 10 \]
Now let's find the decimal approximation of \(\sqrt{97}\). The square root of 97 is approximately:
\[ \sqrt{97} \approx 9.8489 \]
Next, to find the nearest half, we need to round \(9.8489\) to the nearest half. The nearest halves to consider are \(9.5\) and \(10\). Since \(9.8489\) is closer to \(10\) than to \(9.5\), we round it to \(10\).
Thus, \(\sqrt{97}\) is between the whole number 9 and the rational number 10 (when rounded to the nearest half).
In summary:
- The whole number is \(9\).
- The rational number (to the nearest half) is \(10\).
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