To find the approximate value of \( \sqrt{103} \) (103−−−√), we can estimate it using known values on the number line.
We know that:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Since \( 103 \) is between \( 100 \) and \( 121 \), we can conclude that: \[ 10 < \sqrt{103} < 11 \]
Next, we can refine our estimate. Let's check some values:
- \( 10.1^2 = 102.01 \)
- \( 10.2^2 = 104.04 \)
So, we have: \[ 10.1 < \sqrt{103} < 10.2 \]
Now, we can try to find a better approximation:
- \( 10.15^2 = 102.9225 \)
- \( 10.16^2 = 103.2256 \)
This gives us: \[ 10.15 < \sqrt{103} < 10.16 \]
Next, we check \( 10.14 \):
- \( 10.14^2 = 102.8596 \)
This gives us: \[ 10.14 < \sqrt{103} \]
Thus, we can conclude: \[ 10.14 < \sqrt{103} < 10.16 \]
To find a more precise value, we can take an average:
- \( 10.14 \) is approximately \( 10.14 \)
- \( 10.15 \) gives an approximation closer to the middle of the two bounds.
Calculating \( 10.14 \) to two decimal places would yield \( 10.14 \).
However, using a calculator, \( \sqrt{103} \) is approximately \( 10.148\). So rounding to the nearest hundredth, we find:
\[ \sqrt{103} \approx 10.15 \]
Therefore, \( \sqrt{103} \) is approximately \( 10.15 \).
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