This is the 18496th most frequent Russian word.
японскому
"Японскому" means "to Japanese" or "related to Japanese" in the dative case.
In this sentence, "японскому" is in the dative case, agreeing with "языку" to indicate the language being taught.
Я учу японскому языку.
I am teaching the Japanese language.
Here, "японскому" modifies "искусству" (art) in the dative case, showing the art's association with Japanese culture.
Эти книги по японскому искусству.
These books are about Japanese art.
In this example, "японскому" is used in the dative case to describe "экзамену" (exam), indicating its thematic relation to Japanese subjects.