Answer:
[tex]\large\boxed{D.\ 4x^2y\sqrt[4]{3y}}[/tex]
Step-by-step explanation:
[tex]\sqrt[4]{768x^8y^5}\\\\\text{Prime factorization of 768:}\\\\\begin{array}{c|c}768&2\\384&2\\192&2\\96&2\\48&2\\24&2\\12&2\\6&2\\3&3\\1\end{array}\\\\768=2^8\cdot3\\\\\sqrt[4]{768x^8y^5}=\sqrt[4]{(2^8)(3)x^8y^5}\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\\\\=\sqrt{(2^{4+4})(3)x^{4+4}y^{4+1}}=\sqrt{(2^4)(2^4)(3)(x^4)(x^4)(y^4)(y)}\\\\\text{use}\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}[/tex]
[tex]=\sqrt[4]{2^4}\cdot\sqrt[4]{2^4}\cdot\sqrt[4]{x^4}\cdot\sqrt[4]{x^4}\cdot\sqrt[4]{y^4}\cdot\sqrt[4]{3y}\qquad\text{use}\ \sqrt[n]{a^n}=a\\\\=2\cdot2\cdot x\cdot x\cdot y\cdot\sqrt[4]{3y}\\\\=4x^2y\sqrt[4]{3y}[/tex]
