Cubic field extension question

Question

Is the following statement true ?

$$ 2^{\frac{1}{3}} \in \mathbb{Q}(4^{\frac{1}{3}}).$$

Answer 1

Note $(4^{1/3})^2$ is in the field. Write this in another way. And, you should be almost done.

Answer 2

$\Bbb Q(4^{\frac 13}$ has a basis $\{1,4^{\frac 13},4^{\frac 23}\}$. So, $4^{\frac 23}\in \Bbb Q(4^{\frac 13})$. Now, $4^{\frac 23}=2(2^{\frac 13})$ So, $2^{\frac 13}\in \Bbb Q(4^{\frac 13})$[As, $2\in \Bbb Q$].