## Precalculus (6th Edition) Blitzer

The standard form of the expression $3\sqrt{-5}\left( -4\sqrt{-12} \right)$ is $24\sqrt{15}$.
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 $24\sqrt{15}$.