To find the approximate value of the square root of 103, we can start by determining the square roots of nearby perfect squares. The perfect squares around 103 are:
- \(10^2 = 100\)
- \(11^2 = 121\)
Since \(100 < 103 < 121\), we know that:
\[ 10 < \sqrt{103} < 11 \]
Next, to estimate \(\sqrt{103}\) more closely, we can try squaring some decimal values between 10 and 11:
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First, let's try \(10.1\): \[ (10.1)^2 = 102.01 \] This is less than 103.
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Next, let's try \(10.2\): \[ (10.2)^2 = 104.04 \] This is greater than 103.
Now, we know that:
\[ 10.1 < \sqrt{103} < 10.2 \]
To refine our estimate, we can try a value between 10.1 and 10.2, such as \(10.15\):
\[ (10.15)^2 = 103.0225 \] This is slightly greater than 103.
Next, try \(10.14\):
\[ (10.14)^2 = 102.8196 \] This is less than 103.
Now we have:
\[ 10.14 < \sqrt{103} < 10.15 \]
To further narrow it down, we can try \(10.145\):
\[ (10.145)^2 = 103.016025 \] This is also slightly greater than 103.
Next, try \(10.143\):
\[ (10.143)^2 = 102.887249 \] This is less than 103.
Now we approximate further and find that:
\[ 10.143 < \sqrt{103} < 10.145 \]
For final approximation, we will consider that \(\sqrt{103} \approx 10.14\) rounded to the nearest hundredth.
Thus, the approximate value of \(\sqrt{103}\) in decimal form, rounded to the nearest hundredth, is:
\[ \boxed{10.14} \]