This is the 10727th most frequent Russian word.
восьмому
The word "восьмому" is the dative form of "восьмой," meaning "to the eighth" or "for the eighth."
Here, 'восьмому' is used as the dative case form of 'восьмой', meaning 'eighth', referring to 'этажу' (floor).
Я пошёл к восьмому этажу.
I went to the eighth floor.
In this sentence, 'восьмому' is again the dative case form of 'восьмой', correlating with 'уроку' (lesson) indicating "to the eighth".
Мы готовимся к восьмому уроку.
We are preparing for the eighth lesson.
Here, 'восьмому' appears as the dative case of 'восьмой', used specifically to describe 'кругу' (round) and indicating a direction towards it.