The correct option will be : [tex] \frac{\sqrt[4]{24x^3}}{2x} [/tex]
Explanation
[tex] \sqrt[4]{\frac{3}{2x}} = \frac{\sqrt[4]{3}}{\sqrt[4]{2x}} [/tex]
For rationalizing the denominator, first we will multiply both up and down by [tex] \sqrt[4]{2x} [/tex]. So,
[tex] \frac{\sqrt[4]{3}}{\sqrt[4]{2x}} *\frac{\sqrt[4]{2x}}{\sqrt[4]{2x}} \\ \\= \frac{\sqrt[4]{6x}}{\sqrt[4]{4x^2}} [/tex]
Now we will multiply up and down by [tex] \sqrt[4]{4x^2} [/tex]. So, we will get...
[tex] \frac{\sqrt[4]{6x}}{\sqrt[4]{4x^2}} *\frac{\sqrt[4]{4x^2}}{\sqrt[4]{4x^2}} \\ \\ = \frac{\sqrt[4]{24x^3}}{\sqrt[4]{16x^4}} \\ \\ = \frac{\sqrt[4]{24x^3}}{2x} [/tex]
