Question

simplify.
\sqrt{-5} \cdot \sqrt{-10}

Explanation:

Step1: Rewrite using imaginary unit

Recall that \(\sqrt{-a} = i\sqrt{a}\) for \(a>0\). So, \(\sqrt{-5}=i\sqrt{5}\) and \(\sqrt{-10}=i\sqrt{10}\).

Step2: Multiply the two expressions

Multiply \(i\sqrt{5}\) and \(i\sqrt{10}\): \(i\sqrt{5} \cdot i\sqrt{10}=i^2\sqrt{5\times10}\).

Step3: Simplify \(i^2\) and the square root

We know \(i^2 = -1\), and \(5\times10 = 50\), so \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\). Then \(i^2\sqrt{50}=-1\times5\sqrt{2}=-5\sqrt{2}\).

Answer:

\(-5\sqrt{2}\)