To approximate \(\sqrt{103}\), you can consider the perfect squares around 103.
- \(10^2 = 100\)
- \(11^2 = 121\)
Since \(103\) is closer to \(100\) than to \(121\), we can start with an initial guess of \(10.1\) to \(10.2\).
Calculating \(10.1^2\) and \(10.2^2\):
- \(10.1^2 = 102.01\)
- \(10.2^2 = 104.04\)
Since \(102.01 < 103 < 104.04\), we can narrow it down further. Trying \(10.14\):
- \(10.14^2 = 102.6596\)
Trying \(10.15\):
- \(10.15^2 = 103.0225\)
Now, we see that \(10.14^2 < 103 < 10.15^2\), suggesting that \(\sqrt{103}\) is between \(10.14\) and \(10.15\).
For a closer approximation, we can say that:
\[ \sqrt{103} \approx 10.14 \]
Thus, \(\sqrt{103} \approx 10.14\) or approximately \(10.15\). To a suitable degree of accuracy, it's often rounded to \(10.15\).