To approximate the square root of 103, we can start by estimating between the squares of two whole numbers.
We know that:
- \(10^2 = 100\)
- \(11^2 = 121\)
Since \(103\) is between \(100\) and \(121\), we know that:
\[ 10 < \sqrt{103} < 11 \]
Next, we can get closer by testing values between 10 and 11.
Calculating a few decimal values, we find:
- \(10.1^2 = 102.01\)
- \(10.2^2 = 104.04\)
Since \(102.01 < 103 < 104.04\), we know:
\[ 10.1 < \sqrt{103} < 10.2 \]
Let's narrow it down further:
- \(10.15^2 = 103.0225\)
Since \(103.0225 > 103\), we also know:
\[ 10.1 < \sqrt{103} < 10.15 \]
Next, let's test \(10.14\):
- \(10.14^2 = 102.8196\)
Since \(102.8196 < 103\), we have:
\[ 10.14 < \sqrt{103} < 10.15 \]
Thus, we can approximate \(\sqrt{103} \approx 10.14\).
Now, rounding to the nearest hundredth gives us:
\[ \sqrt{103} \approx 10.14 \]