Answer
The standard form of the expression $3\sqrt{-5}\left( -4\sqrt{-12} \right)$ is $24\sqrt{15}$.
Work Step by Step
Consider the expression,$3\sqrt{-5}\left( -4\sqrt{-12} \right)$
Express the square roots of negative numbers in terms of $i$.
$\begin{align}
& 3\sqrt{-5}\left( -4\sqrt{-12} \right)=3i\sqrt{5}\left( -4i\sqrt{12} \right) \\
& =-12{{i}^{2}}\sqrt{5}\left( \sqrt{12} \right)
\end{align}$
Replace the value ${{i}^{2}}=-1$ and make the factors of the radicals.
$\begin{align}
& 3\sqrt{-5}\left( -4\sqrt{-12} \right)=-12\left( -1 \right)\sqrt{5}\left( \sqrt{4\cdot 3} \right) \\
& =12\sqrt{5}\left( 2\sqrt{3} \right) \\
& =24\sqrt{5}\cdot \sqrt{3}
\end{align}$
Use the property $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$.
$\begin{align}
& 3\sqrt{-5}\left( -4\sqrt{-12} \right)=24\sqrt{5\cdot 3} \\
& =24\sqrt{15}
\end{align}$
Therefore, the standard form of the expression $3\sqrt{-5}\left( -4\sqrt{-12} \right)$ is $24\sqrt{15}$.
