To approximate \(\sqrt{103}\), we can find two perfect squares it falls between.
The perfect squares close to 103 are:
- \(10^2 = 100\)
- \(11^2 = 121\)
Thus, \(\sqrt{103}\) is between 10 and 11.
Next, we'll refine our estimate further:
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Estimate between 10 and 11: Check \(10.1\): \[ 10.1^2 = 102.01 \] This is less than 103.
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Check \(10.2\): \[ 10.2^2 = 104.04 \] This is greater than 103.
So, we know \(\sqrt{103}\) is between 10.1 and 10.2.
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Narrow it down further: Check \(10.15\): \[ 10.15^2 = 103.0225 \] This is greater than 103.
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Check \(10.14\): \[ 10.14^2 = 103.0596 \] This is also greater than 103.
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Check \(10.13\): \[ 10.13^2 = 102.6569 \] This is less than 103.
Now we know:
- \(\sqrt{103}\) is between \(10.13\) and \(10.15\).
- Further refine by trying \(10.14\) again with more checks:
- Reiterating for higher precision, let's also check \(10.141\): \[ 10.141^2 = 102.853881 \] This is still less than 103.
Continuing this process lets us get closer:
After checking around the numbers, we determine \(\sqrt{103} \approx 10.14\) to \(10.15\).
Finally: Using a calculator or high precision tables, we find: \[ \sqrt{103} \approx 10.148 \] In decimal form, rounded to the nearest hundredth, this gives us:
\[ \boxed{10.15} \]