To determine which two numbers \( \sqrt{58} \) is between, we can calculate its approximate value.
First, we find a close approximation for \( \sqrt{58} \):
- We know that \( 7^2 = 49 \) and \( 8^2 = 64 \).
- Therefore, \( \sqrt{58} \) is between 7 and 8.
Next, we can refine our estimate by checking the midpoint:
- \( 7.5^2 = 56.25 \) and \( 8^2 = 64 \). Since \( 56.25 < 58 < 64 \), we confirm that \( \sqrt{58} \) is between 7.5 and 8.
Now we'll check \( 7.6 \) and \( 7.7 \):
- \( 7.6^2 = 57.76 \) (this is less than 58).
- \( 7.7^2 = 59.29 \) (this is greater than 58).
Thus, \( \sqrt{58} \) is also between 7.6 and 7.7, but it is already established to be between 7.5 and 8.
By rounding to the nearest half, \( \sqrt{58} \approx 7.62 \), which rounds to 7.5.
In conclusion, the final interval \( \sqrt{58} \) is between is:
between 7.5 and 8.