To determine the whole number and the rational number to the nearest half that \( \sqrt{40} \) (or \( 40^{1/2} \)) is between, we start by calculating \( \sqrt{40} \).
First, we can simplify \( \sqrt{40} \):
\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \]
Now, we need to find the approximate value of \( \sqrt{10} \). The square root of 10 is approximately 3.162. Thus:
\[ \sqrt{40} \approx 2 \times 3.162 \approx 6.324 \]
Now we know that:
\[ \sqrt{40} \approx 6.324 \]
Next, we want to find the whole numbers between which \( 6.324 \) lies. The nearest whole numbers are 6 and 7.
For the rational number to the nearest half, we look at the nearest half-point values around \( 6.324 \). The nearest half is 6.5 but it is not lower than 6. The nearest half below \( 6.324 \) is 6.0.
Thus:
\[ \sqrt{40} \text{ is between } 6 \text{ and } 7 \text{, and } 6.0 \text{ and } 6.5. \]
In summary:
- \( \sqrt{40} \) is between 6 and 7 (whole numbers).
- \( \sqrt{40} \) is between 6.0 and 6.5 (rational numbers to the nearest half).