Describe the separable closure of $\mathbb F_3(u,v)$ in $\mathbb F_3(y,z)$

Question

I'm struggling with the separable closure problem and I don't understand some points.

Let $y,z$ be indeterminates and let $u=y^{12}$ and $v={y^2}{z^{18}}$. Describe the separable closure of $\mathbb F_3(u,v)$ in $\mathbb F_3(y,z)$.

The separable closure contains $y^3$. Then it must contain $z^{54}$ and hence must contain $z^{27}$. Since it must also contain $y^2{z^{18}}$, the separable closure is $\mathbb F_3(y^3, y^2{z^{18}}, z^{27})$. Thus $\mathbb F_3(y,z)$ is totally inseparable over this field.

I don't understand why

1. the separable closure contains $y^3$ and $z^{27}$.
   I guess it is because characteristic is 3. Is it right?

2. the separable closure contains $y^2{z^{18}}$.
   Why "must" the separable closure contain it?