Question

What is the sum of 513−−√+13−−√+213−−√ ?

Answer 1

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 \( \approx 40.85 \).