To find the sum of \( \sqrt{513} + \sqrt{13} + \sqrt{213} \), we start by simplifying each square root as follows:
First, we recognize that:
\[ \sqrt{513} = \sqrt{3 \times 171} = \sqrt{3 \times 3 \times 57} = 3\sqrt{57} \]
Next, we express \( 213 \):
\[ \sqrt{213} = \sqrt{3 \times 71} \]
Thus, the terms can be substituted back into the expression:
\[ \sqrt{513} + \sqrt{13} + \sqrt{213} = 3\sqrt{57} + \sqrt{13} + \sqrt{213} \]
Since \( \sqrt{213} \) does not simplify further, we retain it as is:
Now let's assemble the full sum:
\[ \sqrt{513} + \sqrt{13} + \sqrt{213} = 3\sqrt{57} + \sqrt{13} + \sqrt{213} \]
This is as simplified as possible with straightforward numerical addition.
In numerical terms, we can approximate \( \sqrt{57} \), \( \sqrt{13} \), and \( \sqrt{213} \):
\[ \sqrt{57} \approx 7.55 \ \sqrt{13} \approx 3.61 \ \sqrt{213} \approx 14.59 \]
Calculating these gives:
\[ 3\sqrt{57} \approx 3 \times 7.55 = 22.65 \]
Then summing them gives:
\[ 22.65 + 3.61 + 14.59 \approx 40.85 \]
Thus the final result, approximating as done, gives the overall sum of:
\[ \sqrt{513} + \sqrt{13} + \sqrt{213} \approx 40.85 \]
However, it’s important to note we've provided both an exact algebraic representation and a numerical approximation.
Therefore, the final answer is:
\[ \sqrt{513} + \sqrt{13} + \sqrt{213} \] with an approximate numeric sum being \( \approx 40.85 \).