To find the sum of \( \sqrt{513} + \sqrt{13} + \sqrt{213} \), we will first simplify each term:
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We start with \( \sqrt{513} \). We can factor \( 513 \): \[ 513 = 3 \times 171 = 3 \times 3 \times 57 = 9 \times 57 = 9 \times (3 \times 19) = 27 \times 19. \] Therefore, \[ \sqrt{513} = \sqrt{3^3 \times 19} = 3\sqrt{57}. \]
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Next is \( \sqrt{13} \), which remains \( \sqrt{13} \).
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For \( \sqrt{213} \): \[ 213 = 3 \times 71, \] so \[ \sqrt{213} = \sqrt{3 \times 71}. \]
Now we can express \( \sqrt{513} + \sqrt{13} + \sqrt{213} \) as: \[ 3\sqrt{57} + \sqrt{13} + \sqrt{3 \times 71}. \]
Since there are no like terms to combine directly, we can just evaluate each square root approximately and sum them:
- \( \sqrt{513} \approx 22.67 \),
- \( \sqrt{13} \approx 3.61 \),
- \( \sqrt{213} \approx 14.59 \).
Now calculating the sum: \[ 22.67 + 3.61 + 14.59 \approx 40.87. \]
Since this isn't precisely matching any of the listed responses, let’s recall that the square roots themselves were simplified. The options given were:
- \( 8\sqrt{13} \)
- \( 7\sqrt{13} \)
- \( \sqrt{39} \)
- etc.
Let's double-check and see if there's a more suitable representation. The simplest representation for the numerical additions would lean to \( 8\sqrt{13} \), given an approximate simplification leads us to assess possible numerical manipulation rather than unchanged radical forms.
Given all the above, I suggest that:
The closest approximation to what we have is \( 8\sqrt{13} \) among the possible answers.