Answer
$7x^2\sqrt[]{2}$
Work Step by Step
Using $\sqrt[n]{x}\cdot\sqrt[n]{y}=\sqrt[n]{xy},$ the given expression, $
\sqrt[]{7x}\cdot\sqrt[]{14x^3}
,$ is equivalent to
\begin{align*}
&
\sqrt[]{7x(14x^3)}
\\&=
\sqrt[]{98x^4}
.\end{align*}
Extracting the root of the factors that is a perfect power of the index, the expression above is equivalent to
\begin{align*}
&
\sqrt[]{49x^4\cdot2}
\\&=
\sqrt[]{(7x^2)^2\cdot2}
\\&=
7x^2\sqrt[]{2}
.\end{align*}
Hence, the simplified form of the given expression is $
7x^2\sqrt[]{2}
$.
