Answer
$3a^2\sqrt[4]{108a^2b^2}$
Work Step by Step
Extracting the perfect powers of the index, the given expression, $
3\sqrt[4]{18a^9}\cdot\sqrt[4]{6ab^2}
,$ is equivalent to
\begin{align*}
&
3\sqrt[4]{a^8\cdot18a}\cdot\sqrt[4]{6ab^2}
\\\\&=
3\sqrt[4]{(a^2)^4\cdot18a}\cdot\sqrt[4]{6ab^2}
\\\\&=
3a^2\sqrt[4]{18a}\cdot\sqrt[4]{6ab^2}
.\end{align*}
Using $a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy},$ the expression above is equivalent to
\begin{align*}
&
3a^2\sqrt[4]{18a(6ab^2)}
\\\\&=
3a^2\sqrt[4]{108a^2b^2}
.\end{align*}
Hence, the simplified form of the given expression is $
3a^2\sqrt[4]{108a^2b^2}
$.
