Answer
$6\sqrt[4]{2}$
Work Step by Step
Since $\sqrt{12}$ is equivalent to $\sqrt[4]{12^2},$ the given expression, $
\sqrt[4]{18}\cdot\sqrt[]{12}
,$ is equivalent to
\begin{align*}
&
\sqrt[4]{18}\cdot\sqrt[4]{12^2}
.\end{align*}
Using the laws of exponents, the expression above is equivalent to
\begin{align*}
&
\sqrt[4]{18(12)^2}
&\left( \text{use }\sqrt{a}\cdot \sqrt{b}=\sqrt{ab} \right)
\\\\&=
\sqrt[4]{(9\cdot2)(3\cdot4)^2}
\\\\&=
\sqrt[4]{(3^2\cdot2)(3\cdot2^2)^2}
\\\\&=
\sqrt[4]{(3^2\cdot2)(3^2\cdot2^4)}
&\left( \text{use }(a^x)^y=a^{xy} \right)
\\\\&=
\sqrt[4]{3^{2+2}\cdot2^{1+4}}
&\left( \text{use }a^x\cdot a^y=a^{x+y} \right)
\\\\&=
\sqrt[4]{3^{4}\cdot2^{5}}
\\\\&=
\sqrt[4]{3^{4}\cdot2^{4}\cdot2^1}
\\\\&=
\sqrt[4]{(3\cdot2)^{4}\cdot2}
\\\\&=
(3\cdot2)\sqrt[4]{2}
\\\\&=
6\sqrt[4]{2}
.\end{align*}
Hence, the simplified form of the given expression is $
6\sqrt[4]{2}
.$
