To find the approximate value of \( \sqrt{103} \), we can estimate it using numbers we know.
The square of 10 is \( 100 \) and the square of 11 is \( 121 \). Therefore, \( \sqrt{103} \) is between 10 and 11.
To narrow it down further, we can try some numbers:
- \( 10.1^2 = 102.01 \)
- \( 10.2^2 = 104.04 \)
Since \( 102.01 < 103 < 104.04 \), we know that \( \sqrt{103} \) is between 10.1 and 10.2.
Next, let's test \( 10.15 \):
- \( 10.15^2 = 103.0225 \)
Since \( 10.15^2 \) is slightly more than \( 103 \), we can also check \( 10.14 \):
- \( 10.14^2 = 102.8596 \)
Now we see \( 10.14^2 < 103 < 10.15^2 \), so \( \sqrt{103} \) is between \( 10.14 \) and \( 10.15 \).
Finally, we can try \( 10.14 \) and \( 10.145 \):
- \( 10.145^2 = 102.906025 \) (which is less than 103)
- \( 10.146^2 = 103.021716 \) (which is greater than 103)
This means that
\[ 10.145 < \sqrt{103} < 10.146 \]
Using a calculator, the exact value of \( \sqrt{103} \) is approximately \( 10.148891 \).
Rounding to the nearest hundredth:
\[ \sqrt{103} \approx 10.15 \]
Thus, the approximate value of \( \sqrt{103} \) rounded to the nearest hundredth is \( \boxed{10.15} \).
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