To find the sum \( \sqrt{711} + \sqrt{411} + \sqrt{11} \), we can evaluate each square root separately, then sum them up.
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Calculating \( \sqrt{711} \):
- To approximate \( \sqrt{711} \), we can find two perfect squares that are close to 711. We know \( 26^2 = 676 \) and \( 27^2 = 729 \). This implies that \( 26 < \sqrt{711} < 27 \).
- More precisely, we can use a calculator or approximation technique to find \( \sqrt{711} \approx 26.6 \).
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Calculating \( \sqrt{411} \):
- For \( \sqrt{411} \), we see that \( 20^2 = 400 \) and \( 21^2 = 441 \), so \( 20 < \sqrt{411} < 21 \).
- Using a calculator gives \( \sqrt{411} \approx 20.273 \).
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Calculating \( \sqrt{11} \):
- We know \( 3^2 = 9 \) and \( 4^2 = 16 \), which means \( 3 < \sqrt{11} < 4 \).
- Using a calculator gives \( \sqrt{11} \approx 3.317 \).
Now, we sum these approximate values:
\[ \sqrt{711} + \sqrt{411} + \sqrt{11} \approx 26.6 + 20.273 + 3.317 \]
Calculating the sum:
\[ 26.6 + 20.273 = 46.873 \]
Now adding \( 3.317 \):
\[ 46.873 + 3.317 \approx 50.19 \]
Thus, the sum \( \sqrt{711} + \sqrt{411} + \sqrt{11} \approx 50.19 \).
For an exact final answer, the most suitable approximation is:
\[ \sqrt{711} + \sqrt{411} + \sqrt{11} \approx 50.19 \]
This is a rounded approximation. The exact sum cannot be simplified further without numerical calculations.