My first thought is that $x = a^{1/k}$, so of course $x$ is unique since $a^{1/k}$ is a specific fixed value. But this seemed way too simple of a solution so I think I am misunderstanding the problem.
I am also confused since this appeared in the exercises in a section about Cauchy sequences, but I don't see any relation to Cauchy sequences, whereas all the other problems were about proving something is a Cauchy sequence.
The picture is from the book: Advanced calculus of a single variable by Tunc Geveci, page 42.