Answer
$2xy^2z^2\sqrt[5]{2y^{2}z}$.
Work Step by Step
The given expression is
$\sqrt[5]{8x^4y^3z^3}\cdot \sqrt[5] {8xy^9z^8}$
Apply product rule
$\sqrt[n] a \cdot \sqrt[n] b = \sqrt[n] {ab}$
$\sqrt[5]{8x^4y^3z^3\cdot 8xy^9z^8}$
$\sqrt[5]{2^3x^4y^3z^3\cdot 2^3x^1y^9z^8}$
Use $a^n\cdot a^m = a^{m+n}$
$\sqrt[5]{2^{3+3}x^{4+1}y^{3+9}z^{3+8}}$
$\sqrt[5]{2^{6}x^{5}y^{12}z^{11}}$
Further simplify by using product rule.
$=\sqrt[5]{2^{5+1}x^{5}y^{5+5+2}z^{5+5+1}}$
$=\sqrt[5]{2^{5}2^1x^{5}y^{5}y^{5}y^{2}z^{5}z^{5}z^{1}}$
$=\sqrt[5]{(2xyyzz)^52y^{2}z}$
$=2xyyzz\sqrt[5]{2y^{2}z}$.
$=2xy^2z^2\sqrt[5]{2y^{2}z}$.
