I am trying to show that
free groups (defined via the universal property) are torsion-free.
I have a hint that says that I should first consider words that are cyclically reduced. That is, if $s_1,...,s_k$ are the generators, then $w=s_i...s_l$ is cyclically reduced if $s_i\neq s_l^{-1}$.
Now clearly cyclically reduced words are torsion-free but I am unsure how I can go on to use this to show that all words are torsion-free?
Hint: Every word is the conjugate of a cyclically reduced word. That is, every word $V$ has the form $V\equiv UWU^{-1}$ where $W$ is cyclically reduced and $U$ is a freely reduced word.
(I am going by the hint you have been given, not by your comment about using the universal property. As far as I am concerned, this does not use the universal property of free groups in any explicit way...)