To find the approximate value of \( \sqrt{103} \), we can first consider two perfect squares close to 103. We know that:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Since \( 103 \) lies between \( 100 \) and \( 121 \), we can deduce that:
\[ 10 < \sqrt{103} < 11 \]
Next, we can make an estimation by checking some midpoint values:
- \( 10.1^2 = 102.01 \)
- \( 10.2^2 = 104.04 \)
So we find that:
\[ 10.1 < \sqrt{103} < 10.2 \]
For further approximation, we can check the value of \( 10.15 \):
\[ 10.15^2 = 103.0225 \]
This indicates \( \sqrt{103} < 10.15 \). Now check \( 10.14 \):
\[ 10.14^2 = 102.8196 \]
This suggests \( 10.14 < \sqrt{103} < 10.15 \).
Continuing, let’s check \( 10.14 \):
Continuing the decimal checks, we can deduce that:
\[ \sqrt{103} \approx 10.14 \]
Now calculating directly using a calculator, we find:
\[ \sqrt{103} \approx 10.148 \]
Rounded to the nearest hundredth, the approximate value of \( \sqrt{103} \) is:
\[ \sqrt{103} \approx 10.15 \]
Thus, the answer is:
\[ \sqrt{103} \text{ is approximately } 10.15. \]
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