Given expression: [tex]\sqrt[4]{\frac{24x^6y}{128x^4y^5} }[/tex].
[tex]\mathrm{Cancel\:the\:common\:factor:}\:8[/tex]
[tex]=\frac{3x^6y}{16x^4y^5}[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}\:=\:x^{a-b}[/tex]
[tex]\frac{x^6}{x^4}=x^{6-4}=x^2[/tex]
[tex]=\frac{3x^2y}{16y^5}[/tex]
[tex]\mathrm{Cancel\:the\:common\:factor:}\:y[/tex]
[tex]=\frac{3x^2}{16y^4}[/tex]
[tex]=\sqrt[4]{\frac{3x^2}{16y^4}}[/tex]
[tex]\mathrm{Apply\:radical\:rule\:}\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}[/tex]
[tex]=\frac{\sqrt[4]{3x^2}}{\sqrt[4]{16y^4}}[/tex]
[tex]=\frac{\sqrt[4]{3x^2}}{\sqrt[4]{16}\sqrt[4]{y^4}}[/tex]
[tex]=\frac{\sqrt[4]{3}\sqrt[4]{x^2}}{\sqrt[4]{16}\sqrt[4]{y^4}}[/tex]
[tex]=\frac{\sqrt[4]{3}\sqrt[4]{x^2}}{2\sqrt[4]{y^4}}[/tex]
[tex]=\frac{\sqrt[4]{3x^2}}{2y}[/tex].
Therefore, correct option is 4th option [tex]\frac{\sqrt[4]{3x^2}}{2y}[/tex].
