What are the irreducible components of $V(xy-z^3,xz-y^3)$ in $\mathbb{A}^3_K$?
Let $(x,y,z) \in V(xy-z^3,xz-y^3)$. Two cases:
$1$. $x=0$. Then $y=z=0$.
$2$. $x\ne 0$. Then $y^4 = z^4$, so $z = \omega \cdot y$ where $\omega^4 = 1$. If $y=0$ then $z=0$. If $y \ne 0$ then $z \ne 0$ and we have $(x,y,z) = (\omega^3 y^2, y, \omega y)$.
Therefore we have one component, the line $y=z=0$, and for each $\omega$, a fourth root of $1$, we have a component isomorphic to $\mathbb{A}^1$: $\{ (\omega^3 t^2, t, \omega t )\ | \ t \in k \}$.