Answer
$$4x^2$$
Work Step by Step
RECALL:
(1) For $a\gt 0, b \gt0$, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.
(2) If $n$ is odd, $\sqrt[n]{a^n} = a$.
(3) If $n$ is even, $\sqrt[n]{a^n} = |a|$.
Using rule (1) above gives:
\begin{align*}
\sqrt{8x^2} \cdot \sqrt{2x^2}&=\sqrt{8x^2(2x^2)}\\
&=\sqrt{16x^4}\\
&=\sqrt{(4x^2)(4x^2)}\\
&=\sqrt{\left(4x^2\right)^2}\\
\end{align*}
Use rule (3) above to obtain:
\begin{align*}
\sqrt{\left(4x^2\right)^2}&=|4x^2|\\
\end{align*}
However, since the value of $x^2$ is never negative, then $|x^2|=x^2$.
Thus,
\begin{align*}
|4x^2|=4x^2
\end{align*}
