To find the approximate value of \( \sqrt{103} \), we can first identify two perfect squares it lies between. We know:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Thus, \( \sqrt{103} \) is between 10 and 11.
To narrow it down further, we can check the midpoints:
- \( 10.1^2 = 102.01 \)
- \( 10.2^2 = 104.04 \)
Since \( 102.01 < 103 < 104.04 \), we know \( \sqrt{103} \) is between 10.1 and 10.2.
Next, let's try \( 10.15 \) to see if it’s closer:
- \( 10.15^2 = 103.0225 \)
This value is greater than 103. Thus, we now know that:
- \( 10.1 < \sqrt{103} < 10.15 \)
Let’s check \( 10.14 \):
- \( 10.14^2 = 102.8596 \)
This value is less than 103. Therefore, we have:
- \( 10.14 < \sqrt{103} < 10.15 \)
To refine it further, we can look at \( 10.141 \):
- \( 10.141^2 \approx 102.844281 \) (less than 103)
- \( 10.142^2 \approx 102.846564 \) (still less than 103)
- \( 10.143^2 \approx 102.848849 \) (still less than 103)
- \( 10.144^2 \approx 102.851136 \) (still less than 103)
- \( 10.145^2 \approx 102.853225 \) (still less than 103)
- \( 10.146^2 \approx 102.855616 \) (less than 103)
Eventually, we find:
- \( 10.14 < \sqrt{103} < 10.146 \)
Continuing this process, through either formal calculations or using a calculator, we find:
\[ \sqrt{103} \approx 10.148 \]
Finally, rounding this to the nearest hundredth gives:
\[ \sqrt{103} \approx 10.15 \]
Thus, the approximate value of \( \sqrt{103} \) is \(\boxed{10.15}\).
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