Question

Using a number line with whole number and rational number to the nearest half is square root 97 between?

Answer 1

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 \(10\).