Answer
$=2xyz\sqrt[4]{x^{2}z^{3}}$.
Work Step by Step
The given expression is
$\sqrt[4]{8x^2y^3z^6}\cdot \sqrt[4] {2x^4yz}$
Apply product rule
$\sqrt[n] a \cdot \sqrt[n] b = \sqrt[n] {ab}$
$\sqrt[4]{8x^2y^3z^6\cdot 2x^4yz}$
$\sqrt[4]{2^3x^2y^3z^6\cdot 2^1x^4y^1z^1}$
Use $a^n\cdot a^m = a^{m+n}$
$\sqrt[4]{2^{3+1}x^{2+4}y^{3+1}z^{6+1}}$
$\sqrt[4]{2^{4}x^{6}y^{4}z^{7}}$
Further simplify by using product rule.
$=\sqrt[4]{2^{4}x^{4+2}y^{4}z^{4+3}}$
$=\sqrt[4]{2^{4}x^{4}x^{2}y^{4}z^{4}z^{3}}$
$=\sqrt[4]{(2xyz)^4x^{2}z^{3}}$
$=2xyz\sqrt[4]{x^{2}z^{3}}$.
