Answer
$12a^2\sqrt{b}$
Work Step by Step
Using $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab},$ the given expression, $
\sqrt{12a}\sqrt{12a^3b}
,$ is equivalent to
\begin{align*}
&
\sqrt{12a(12a^3b)}
\\&=
\sqrt{12^2a^4b}
.\end{align*}
Extracting the roots of the factors that are perfect powers of the index, the expression above is equivalent to
\begin{align*}\require{cancel}
&
\sqrt{(12a^2)^2b}
\\\\&=
|12a^2|\sqrt{b}
&(\text{use }\sqrt{x^2}=|x|)
\\&=
12a^2\sqrt{b}
&(\text{since }|12|=12,\text{ }|a^2|=a^2)
.\end{align*}
Hence, the expression $\sqrt{12a}\sqrt{12a^3b}$ simplifies to $12a^2\sqrt{b}$.
