To find the square root of 103 (written as \( \sqrt{103} \)), we can start by determining its approximate value. We know that:
- \( \sqrt{100} = 10 \)
- \( \sqrt{121} = 11 \)
Since 103 is between 100 and 121, we know that:
\[ 10 < \sqrt{103} < 11 \]
Now, let's find a better approximation by performing some calculations:
- Estimate closer to 10:
- \( 10.1^2 = 102.01 \)
- \( 10.2^2 = 104.04 \)
This indicates that \( \sqrt{103} \) is between 10.1 and 10.2.
- Further refine:
- Check \( 10.15^2 = 103.0225 \)
- Check \( 10.14^2 = 102.6596 \)
Now, we see that \( 10.14^2 < 103 < 10.15^2 \), suggesting a more precise approximation.
So, \( \sqrt{103} \approx 10.14 \).
Finally, rounding 10.14 to the nearest tenth gives us:
\[ \sqrt{103} \approx 10.1 \]
Thus, \( \sqrt{103} \) is approximately 10.1 when rounded to the nearest tenth.