Question

simplify.
\sqrt{-6} \cdot \sqrt{-3}

Explanation:

Step1: Rewrite using imaginary unit

Recall that \(\sqrt{-a}=i\sqrt{a}\) for \(a>0\). So, \(\sqrt{-6}=i\sqrt{6}\) and \(\sqrt{-3}=i\sqrt{3}\).

Step2: Multiply the expressions

Multiply \(i\sqrt{6}\) and \(i\sqrt{3}\): \((i\sqrt{6})\cdot(i\sqrt{3}) = i^2\cdot\sqrt{6\cdot3}\).

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

We know \(i^2 = - 1\), and \(\sqrt{6\cdot3}=\sqrt{18}=\sqrt{9\cdot2}=3\sqrt{2}\). Substitute these values: \(i^2\cdot\sqrt{18}=-1\cdot3\sqrt{2}=-3\sqrt{2}\).

Answer:

\(-3\sqrt{2}\)