To determine the range of numbers that \(\sqrt{10}\) (which is \(10^{0.5}\)) falls between, we first need to calculate its approximate value.
The square root of 10 is approximately:
\[ \sqrt{10} \approx 3.162 \]
Next, we want to find the whole numbers it is between:
- The whole number just below \(\sqrt{10}\) is 3.
- The whole number just above \(\sqrt{10}\) is 4.
Thus, we can say:
\[ 3 < \sqrt{10} < 4 \]
Next, we determine the nearest half to \(\sqrt{10}\):
- The nearest halves around 3.162 are 3.0 and 3.5. Since 3.162 is closer to 3.0 than to 3.5, we can use these values as well.
Consequently, we find that:
\[ \sqrt{10} \text{ is between } 3 \text{ and } 4, \text{ and between } 3.0 \text{ and } 3.5. \]
Therefore, your final answer is:
\(\sqrt{10}\) is between \(3\) and \(4\), and between \(3.0\) and \(3.5\).