Square-free numbers of the form $\lfloor m^c\rfloor$ with smooth indices

The aim of this article was to study the square-free numbers of the form $\lfloor m^c\rfloor$ with smooth indices, which can be seen as an extension of a result of Cao and Zhai. As an extension to the work of Cao and Zhai, the proof of Theorem 1.1 employed the division trick, which transformed an arithmetic problem into a problem concerning exponential sum estimates. The division trick proved to be more direct and concise in comparison with alternative approaches. For the sake of completeness of our results, Theorem 1.2, serving as a complement to Theorem 1.1, can be proved by invoking results from the work of Cao and Zhai. Combining Theorems 1.1 and 1.2, we obtained our full conclusion, namely, Theorem 1.3.