Periodicity of words

Question

If we have a non-periodic word $u$. Is it possible to have another word $\beta = \gamma u^{i-1}$ with $i>1$ and $\gamma \neq u^*$ so that $\beta$ is periodic. it's intuitive to say that it is not possible, but I don't know how to prove or disprove the statement.

Answer 1

Let the alphabet be the set $\{x,y\}$.

Let $u = xy$, let $i=2$, and let $\gamma=xxyx$.

Then $u$ is not periodic, but $$\beta=\gamma u^{i-1} = (xxyx)(xy) = (xxy)(xxy)$$ which is periodic.