We rationalize fractions in order to remove square roots from the denominator. To do so, we must multiply the numerator and denominator by the same value.
In this case, we can multiply by [tex] \sqrt[3]{4^2} [/tex] which will cancel out the cube root, leaving the denominator as 4.
[tex] \frac{ \sqrt[3]{2} }{ \sqrt[3]{4} } \times \frac{ \sqrt[3]{16} }{ \sqrt[3]{16} } [/tex]
The result is:
[tex] \frac{ \sqrt[3]{32} }{4} [/tex]
This can be reduced by pulling an 8 out from the radicand:
[tex] \frac{2 \sqrt[3]{4} }{4} [/tex]
Furthermore:
[tex] \frac{ \sqrt[3]{4} }{2} [/tex]
The last option is the correct answer.
