To multiply \(\sqrt{519}\) and \(\sqrt{62}\), we can use the property of square roots that states \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
So, we have:
\[ \sqrt{519} \cdot \sqrt{62} = \sqrt{519 \cdot 62} \]
Now, let's calculate \(519 \cdot 62\):
\[ 519 \cdot 62 = 32178 \]
Therefore,
\[ \sqrt{519} \cdot \sqrt{62} = \sqrt{32178} \]
Thus, the exact answer is
\[ \sqrt{32178} \]
If we want to simplify \(\sqrt{32178}\), we can factor 32178 to see if it has a perfect square factor:
- First, we will check for divisibility by small prime numbers.
- After calculating, we find that \(32178 = 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 13\).
Thus, we can simplify:
\[ \sqrt{32178} = \sqrt{2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 13} = 3 \cdot \sqrt{2 \cdot 7 \cdot 11 \cdot 13} = 3 \cdot \sqrt{2002} \]
So the fully simplified exact answer is:
\[ 3 \sqrt{2002} \]
3 answers