Answer
$2xy^2z\sqrt[5]{2y^{3}z}$.
Work Step by Step
The given expression is
$\sqrt[5]{8x^4y^6z^2}\cdot \sqrt[5] {8xy^7z^4}$
Apply product rule
$\sqrt[n] a \cdot \sqrt[n] b = \sqrt[n] {ab}$
$\sqrt[5]{8x^4y^6z^2\cdot 8xy^7z^4}$
$\sqrt[5]{2^3x^4y^6z^2\cdot 2^3x^1y^7z^4}$
Use $a^n\cdot a^m = a^{m+n}$
$\sqrt[5]{2^{3+3}x^{4+1}y^{6+7}z^{2+4}}$
$\sqrt[5]{2^{6}x^{5}y^{13}z^{6}}$
Further simplify by using product rule.
$=\sqrt[5]{2^{5+1}x^{5}y^{5+5+3}z^{5+1}}$
$=\sqrt[5]{2^{5}2^1x^{5}y^{5}y^{5}y^{3}z^{5}z^{1}}$
$=\sqrt[5]{(2xyyz)^52y^{3}z}$
$=2xyyz\sqrt[5]{2y^{3}z}$.
$=2xy^2z\sqrt[5]{2y^{3}z}$.
