I was going to ask another question about this - Origin - Elementary Number Theory, Jones, p23, Lemma 2.4 - but then I chanced on If a product of relatively prime integers is an $n$th power, then each is an $n$th power and its duplicate. I'm bad at searching and I don't want to instigate more duplicates, thence I want to check if there are any others on this before asking newly.
Is there a name for this Lemma so I can try to search for it here and on Google? If you find anything, please inform me exactly what you typed in.
If $a_1,\dots,a_r$ are mutually coprime positive integers, and $a_1\dots a_r$ is an $m$-th power of some integer $m\ge2$, then each $a_i$ is an $m$-th power.