To find which whole number and rational number to the nearest half \( \sqrt{68} \) is between, we will first calculate \( \sqrt{68} \).
-
Estimate \( \sqrt{68} \):
\( \sqrt{68} \) can be simplified: \[ \sqrt{68} = \sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17} \] Now, we'll estimate \( \sqrt{17} \). Since \( 4^2 = 16 \) and \( 5^2 = 25 \), we know: \[ 4 < \sqrt{17} < 5 \] Hence, \( 2\sqrt{17} \) would be: \[ 8 < 2\sqrt{17} < 10 \] -
Narrowing Down \( \sqrt{17} \):
To get a better approximation, \( \sqrt{17} \) is approximately 4.123 (calculated through a calculator or further approximation). Thus: \[ 2\sqrt{17} \approx 2 \times 4.123 \approx 8.246 \] -
Determine between which whole numbers and rational numbers \( \sqrt{68} \) lies:
- The whole number part of \( \sqrt{68} \approx 8.246 \) indicates it lies between 8 and 9.
- To find the rational number to the nearest half, evaluate:
- The nearest half value around \( 8.246 \) is \( 8.5 \).
- Therefore, \( \sqrt{68} \) is between 8 and 8.5 when rounded to the nearest half.
In conclusion, \( \sqrt{68} \) is between 8 and 8.5.
Final Answer:
between 8 and 8.5.