How do I find the value of $operatorname{LCM} (a_1^k,a_2^k, ldots ,a_n^k) pmod {m}$?

Hint: there is no need to actually calculate $LCM(a_1,ldots,a_n)$. We only have to find $LCM bmod m$. Instead calculate $d=GCD(a_1,ldots,a_n,m)$.

Let $x = LCM(a_1,ldots,a_n) bmod m$. This means that $x = LCM - ell m$ for some integer $ell$. The right hand side is divisible by $d$, therefore $x$ is divisible by $d$ as well.
So we have $x = dcdot(frac{LCM}d - ell frac md)$ and we can write: $$x=dcdotleft(frac{LCM}{d} bmod frac mdright) = dcdotleft(LCM(frac{a_1}d,ldots,frac{a_n}d)bmod frac mdright).$$