It is an old result of Schützenberger that in a free group, a basic commutator cannot be a proper power. A look at the original reference

M.-P. Schützenberger, Sur l'équation $a^{2+n} = b^{2+m}c^{2+p}$ dans un groupe libre, C. R. Acad. Sci. Paris 248 (1959), 2435–2436 (French).

quickly reveals that a lot of details are missing and some claims appear to be wrong.

Question: Let $F$ be a free group and $a,b,c \in F$ with $c \neq 1$ and $n \geq 2$. Why is $[a,b] \neq c^n$?

In particular, it would be nice to have a somewhat geometric proof of this apparently fundamental fact. There is an algebraic proof in

G. Baumslag, Some aspects of groups with unique roots, Acta Math 104(3) (1960), 217–303.

as Lemma 36.4 but it is relies on various technical computations and is hard to grasp.