To determine the values that \(\sqrt{40}\) is between, we start by calculating \(\sqrt{40}\).
-
Estimate \(\sqrt{40}\):
- We know that \(6^2 = 36\) and \(7^2 = 49\), so: \[ 6 < \sqrt{40} < 7 \]
-
Find a more precise decimal value:
-
We can simplify \(\sqrt{40}\): \[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10} \]
-
The approximate value of \(\sqrt{10}\) is about \(3.162\), so: \[ \sqrt{40} \approx 2 \times 3.162 \approx 6.324 \]
-
-
Rounding to the nearest half:
- The number \(6.324\) is closer to \(6.5\) than to \(6.0\), so: \[ \sqrt{40} \approx 6.5 \text{ (to the nearest half)} \]
-
Determine the whole numbers and rational numbers:
- The whole number it is between is \(6\) and \(7\).
- The rational number to the nearest half is \(6.5\).
Thus, the final answer is: \[ \sqrt{40} \text{ is between } 6 \text{ and } 7, \text{ and the rational number to the nearest half is } 6.5. \]