Answer
$=2xyz^2\sqrt[4]{2x^{2}z}$.
Work Step by Step
The given expression is
$\sqrt[4]{4x^2y^3z^3}\cdot \sqrt[4] {8x^4yz^6}$
Apply product rule
$\sqrt[n] a \cdot \sqrt[n] b = \sqrt[n] {ab}$
$\sqrt[4]{4x^2y^3z^3\cdot 8x^4yz^6}$
$\sqrt[4]{2^2x^2y^3z^3\cdot 2^3x^4y^1z^6}$
Use $a^n\cdot a^m = a^{m+n}$
$\sqrt[4]{2^{2+3}x^{2+4}y^{3+1}z^{3+6}}$
$\sqrt[4]{2^{5}x^{6}y^{4}z^{9}}$
Further simplify by using product rule.
$=\sqrt[4]{2^{4+1}x^{4+2}y^{4}z^{4+4+1}}$
$=\sqrt[4]{2^{4}2x^{4}x^{2}y^{4}z^{4}z^{4}z^1}$
$=\sqrt[4]{(2xyzz)^42x^{2}z}$
$=2xyzz\sqrt[4]{2x^{2}z}$.
$=2xyz^2\sqrt[4]{2x^{2}z}$.
