To approximate the value of \(\sqrt{103}\) using a number line, we can start by identifying the perfect squares around 103:
- \(10^2 = 100\)
- \(11^2 = 121\)
Since \(103\) is between \(100\) and \(121\), we know that:
\[ 10 < \sqrt{103} < 11 \]
Next, we can determine that \(\sqrt{103}\) is closer to \(10\) than to \(11\). To find a more precise approximation, we can try squaring numbers between 10 and 11.
Calculating \(10.1^2\):
\[ 10.1^2 = 102.01 \]
Calculating \(10.2^2\):
\[ 10.2^2 = 104.04 \]
Since \(102.01 < 103 < 104.04\), we can narrow down \(\sqrt{103}\) to be between \(10.1\) and \(10.2\).
To get even closer, we can calculate \(10.11^2\):
\[ 10.11^2 \approx 102.2121 \]
Calculating \(10.12^2\):
\[ 10.12^2 \approx 102.4144 \]
Calculating \(10.13^2\):
\[ 10.13^2 \approx 102.6169 \]
And \(10.14^2\):
\[ 10.14^2 = 102.6196 \]
When you calculate:
\[ 10.14 < \sqrt{103} < 10.15 \]
However, using a calculator, the square root value is approximately:
\[ \sqrt{103} \approx 10.148 \]
Rounding \(10.148\) to the nearest hundredth gives:
\[ \sqrt{103} \approx 10.15 \]
Therefore, rounding to the nearest hundredth, the approximate value of \(\sqrt{103}\) is:
\[ \sqrt{103} \approx 10.15 \]
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