To multiply the two square roots \( \sqrt{519} \) and \( \sqrt{62} \), you can use the property of square roots that states:
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \]
Applying this property:
\[ \sqrt{519} \cdot \sqrt{62} = \sqrt{519 \cdot 62} \]
Now, we need to calculate \( 519 \cdot 62 \):
\[ 519 \cdot 62 = 32178 \]
Thus, we have:
\[ \sqrt{519} \cdot \sqrt{62} = \sqrt{32178} \]
Therefore, the exact answer is:
\[ \sqrt{32178} \]