A formation $\mathfrak F$ is called radical (weakly $n$-radical) if it contains every group $G$ which can be represented in the form $G=M_1M_2\dots M_n$, $M_i\triangleleft G$, where the subgroups $M_i$ belong to $\mathfrak F$ (belong to $\mathfrak F$ and have pairwise prime indices). It is proved that a local formation $\mathfrak F$ is radical (weakly $n$-radical, $n\ge2$) if and only if its complete inner local screen $f$ has the following property: $f(p)$ is a radical (a weakly $n$-radical, $n\ge2$) formation for every prime number $p$.