Multiplying primitive roots of unity.

1. The problem statement, all variables and given/known data





Let ##ζ_3## and ##ζ_5## denote the 3rd and 5th primitive roots of unity respectively. I was wondering if I could write the product of these in the form ##ζ_n^k## for some n and k.





2. Relevant equations







3. The attempt at a solution







We know that ##ζ_3## is a root of ##x^3=1##, and ##ζ_5## is a root of ##x^5=1##, so ##ζ_3ζ_5## must be a root of ##x^{15}=1##, right?...so ##ζ_3ζ_5 = ζ_{15}##. And in general ##ζ_nζ_k = ζ_{[n,k]}##, where [n,k] denotes the lcm of n and k. Is that right?



Also is it true that ##ζ_{15}^8 = ζ_{15}##? If so, then why/how?



Thank you in advance

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