To find the value of \( \sqrt{105} \), we can estimate its value.
Since \( 10^2 = 100 \) and \( 11^2 = 121 \), we know that:
\[ 10 < \sqrt{105} < 11 \]
Next, we can refine our estimate a bit more. To narrow it down, we can calculate:
\[ 10.5^2 = 110.25 \]
Since \( 110.25 > 105 \), we know:
\[ \sqrt{105} < 10.5 \]
Now let's check \( 10.25 \):
\[ 10.25^2 = 105.0625 \]
Since \( 105.0625 > 105 \), we conclude that:
\[ \sqrt{105} < 10.25 \]
Now checking \( 10.2 \):
\[ 10.2^2 = 104.04 \]
Since \( 104.04 < 105 \), we have:
\[ 10.2 < \sqrt{105} \]
Thus, the range we have is:
\[ 10.2 < \sqrt{105} < 10.25 \]
This means \( \sqrt{105} \) is between 10 and 10.5.
The final answer is:
between 10 and 10.5.