To approximate the square root of 103, we can find perfect squares close to 103.
We know that:
- \(10^2 = 100\)
- \(11^2 = 121\)
This tells us that \(10 < \sqrt{103} < 11\).
Next, we will refine our approximation by checking some decimal values between 10 and 11.
Calculating:
- \(10.1^2 = 102.01\)
- \(10.2^2 = 104.04\)
This indicates that:
- \(10.1^2 < 103 < 10.2^2\), so \(10.1 < \sqrt{103} < 10.2\).
We can further refine this by checking halfway, say \(10.15\):
- \(10.15^2 = 103.0225\)
Now, we see that:
- \(10.1 < \sqrt{103} < 10.15\).
Let's check \(10.14\):
- \(10.14^2 = 102.6596\)
Now, we have:
- \(10.14 < \sqrt{103} < 10.15\).
Since \(10.14\) is close, we can conclude: The approximate value of \(\sqrt{103} \approx 10.14\).
Rounding to the nearest hundredth, we find: \[ \sqrt{103} \approx 10.14 \]
Therefore, the answer is \( \boxed{10.14} \).
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