To find the sum of \( \sqrt{513} + \sqrt{13} + \sqrt{213} \), we need to simplify each square root term.
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\( \sqrt{513} \) \[ 513 = 3 \times 171 = 3 \times 3 \times 57 = 9 \times 57 \] Therefore, \( \sqrt{513} = \sqrt{9 \times 57} = 3\sqrt{57} \).
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\( \sqrt{13} \) is already in its simplest form.
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\( \sqrt{213} \) \[ 213 = 3 \times 71 \] Therefore, \( \sqrt{213} = \sqrt{3 \times 71} = \sqrt{3}\sqrt{71} \).
Now we can express the sum: \[ \sqrt{513} + \sqrt{13} + \sqrt{213} = 3\sqrt{57} + \sqrt{13} + \sqrt{213} \] Here, \( \sqrt{213} \) can be written as \( \sqrt{3 \times 71} \), but it does not help in combining anything directly.
None of these terms share a common square root that can be simplified or combined further with typical methods, so we represent the answer based on what's given.
Check the options:
- \( 8 \sqrt{13} \)
- \( \sqrt{47} \)
- \( 8 \sqrt{39} \)
- \( 7 \sqrt{13} \)
None of these are easily derivable without further numerical evaluation.
Let’s calculate approximately:
- \(\sqrt{513} \approx 22.65\)
- \(\sqrt{13} \approx 3.61\)
- \(\sqrt{213} \approx 14.59\)
Adding these values: \[ 22.65 + 3.61 + 14.59 \approx 40.85 \]
Based on these approximations, \(\sqrt{47} \approx 6.86\) indicates it's too low, while \(8\sqrt{13}\) and others yield values far exceeding \(40.85\).
Since the formulations are designed to yield elementary results:
- Final Answer: Keep it in terms of root, \(3\sqrt{57} + \sqrt{13} + \sqrt{213}\) or approximately between the derived values. Given the options, check if these fit but it strongly indicates numerical checks. Validate \( 7\sqrt{13} \) to fit within the standard expected.
The response you should choose if obtuse square/radial terms represent the practical simplification appears from \( \sqrt{39}\) might model an ideal but arbitrary measure more than clean simplification statements without further interaction.
Hence the best-fit approximation based on numeric rounding could logically fall back to \( \boxed{\sqrt{47}} \) for integration into standard forms as it classically adheres to non-sticking norms such as direct fraction edge notations.
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