To estimate \( \sqrt{53} \) more accurately between the suggested values, we can calculate a few close squares and then narrow it down.
We know:
- \( 7.2^2 = 51.84 \)
- \( 7.3^2 = 53.29 \)
Since \( 51.84 < 53 < 53.29 \), \( \sqrt{53} \) is indeed between 7.2 and 7.3.
Next, we can further estimate \( \sqrt{53} \) by checking \((7.25^2)\):
\[ 7.25^2 = 52.5625 \]
Now we see:
- \( 7.2^2 = 51.84 \)
- \( 7.25^2 = 52.5625 \)
- \( 7.3^2 = 53.29 \)
Since \( 52.5625 < 53 < 53.29 \), \( \sqrt{53} \) is between 7.25 and 7.3.
Next, we check \( (7.27^2) \):
\[ 7.27^2 = 52.8129 \]
Now we see:
- \( 7.25^2 = 52.5625 \)
- \( 7.27^2 = 52.8129 \)
- \( 7.3^2 = 53.29 \)
Since \( 52.8129 < 53 < 53.29 \), \( \sqrt{53} \) is between 7.27 and 7.3.
Continuing, we check \( (7.28^2) \):
\[ 7.28^2 = 52.9984 \]
Now we see:
- \( 7.27^2 = 52.8129 \)
- \( 7.28^2 = 52.9984 \)
- \( 7.3^2 = 53.29 \)
Since \( 52.9984 < 53 < 53.29 \), \( \sqrt{53} \) is between 7.28 and 7.3.
Finally, we check \( (7.29^2) \):
\[ 7.29^2 = 53.0641 \]
Now we see:
- \( 7.28^2 = 52.9984 \)
- \( 7.29^2 = 53.0641 \)
Since \( 52.9984 < 53 < 53.0641 \), we find that \( \sqrt{53} \) lies between \( 7.28 \) and \( 7.29 \).
Thus, the correct answer is:
7.28 and 7.29.