To find where \( \sqrt{58} \) lies, we first calculate its approximate value.
Calculating \( \sqrt{58} \):
- Finding perfect squares around 58:
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
Since \( 49 < 58 < 64 \), we know that \( 7 < \sqrt{58} < 8 \).
- Getting a more precise estimate:
- We can check \( 7.5 \): \[ 7.5^2 = 56.25 \]
- Check \( 7.6 \): \[ 7.6^2 = 57.76 \]
- Check \( 7.7 \): \[ 7.7^2 = 59.29 \]
Since \( 56.25 < 58 < 57.76 \), we narrow it down to \( 7.5 < \sqrt{58} < 7.6 \).
So, based on these calculations, we conclude that:
- The whole number part is 7.
- The rational number to the nearest half is 7.5.
Final placement on the number line:
- \( \sqrt{58} \) is between 7.5 and 8.