Question

look at the example below which shows how the product property of radicals is used to simplify a radical.
\\(\sqrt{-36} = \sqrt{36}\sqrt{-1} = 6i\\)
use the product property of radicals to simplify the following radical.
\\(\sqrt{-81} = \\) \\(\sqrt{9i}\\)
complete
simplify \\(\sqrt{-50}\\).
\\(\circ\\) \\(5\sqrt{2}\\)
\\(\circ\\) \\(5i\sqrt{2}\\)
\\(\circ\\) \\(-5\sqrt{2}\\)
\\(\circ\\) \\(-5i\sqrt{2}\\)
done

Explanation:

Step1: Apply product property of radicals

For \(\sqrt{-50}\), we can write it as \(\sqrt{50\times(-1)}\). By the product property of radicals \(\sqrt{ab}=\sqrt{a}\sqrt{b}\) (for \(a\geq0,b\geq0\), here we extend it for complex numbers where we consider \(\sqrt{-1}=i\)), so \(\sqrt{-50}=\sqrt{50}\times\sqrt{-1}\).

Step2: Simplify \(\sqrt{50}\)

We factorize \(50 = 25\times2\), so \(\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\) (since \(\sqrt{25} = 5\)).

Step3: Substitute \(\sqrt{-1}=i\)

We know that \(\sqrt{-1}=i\), so substituting back, we get \(\sqrt{-50}=5\sqrt{2}\times i = 5i\sqrt{2}\).

Answer:

\(5i\sqrt{2}\) (corresponding to the option "5\(i\)\(\sqrt{2}\)")