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.