To find the whole number and the rational number to the nearest half that \( \sqrt{10} \) (or \( 10^{1/2} \)) is between, we first need to calculate the approximate value of \( \sqrt{10} \).
Calculating \( \sqrt{10} \):
- We know that \( 3^2 = 9 \) and \( 4^2 = 16 \), so \( \sqrt{10} \) is between 3 and 4.
- To get a more precise estimate, we can calculate \( \sqrt{10} \) using a calculator or estimating. It is approximately \( 3.16 \).
Now, we need to find the nearest half:
- The nearest halves around \( 3.16 \) are \( 3.0 \) and \( 3.5 \).
Thus, rounding \( \sqrt{10} \) to the nearest half gives us:
- \( 3.0 \) (the whole number)
- \( 3.0 \) (the rational number to the nearest half)
In conclusion, \( \sqrt{10} \) is between the whole number 3 and the rational number 3.0 (nearest half).