  A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if
the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many
values in a group $G\in \mathcal X$. It is a long-standing open problem whether
every word is concise in residually finite groups. In this paper we observe
that the conciseness of a word $w$ in residually finite groups is equivalent to
that in the class of virtually pro-$p$ groups.
  This is used to show that if $q,n$ are positive integers and $w$ is a
multilinear commutator word, then the words $w^q$ and $[w^q,_{n} y]$ are
concise in residually finite groups. Earlier this was known only in the case
where $q$ is a prime power.
  In the course of the proof we establish that certain classes of groups
satisfying the law $w^q\equiv1$, or $[\delta_k^q,{}_n\, y]\equiv1$, are
varieties.
