Let $m \in \Bbb{Z}_p$ be fixed. Let $a_1,...a_l$ be fixed integers. I am trying to find out solutions of the equation $m=x_1^{a_1}...x_l^{a_l}$ where $x_1,...,x_l\in \Bbb{Z}_p$. Here $x_1,...x_l$ are variables.
Is it possible to find at least one solution of this equation? Can we find at least one solution if we vary $l$?
No, not always. For example take $\mathbb{Z}_{p}$ where $p$ is odd, let $m$ be a nonsquare, and $a_{1}=\cdots=a_{l} =2$. Then there is no solution to $$m = x_{1}^{2} \cdots x_{l}^{2}$$ because $m$ is nonsquare and a product of squares is a square. This is not affected if you vary $l$.
Actually it makes no sense to allow $l$ to vary, since you could always put an $x_{i} = 1$ and then the corresponding $a_{i}$ is essentially excluded from the calculation.
EDIT I don't have a proof at my fingertips, but I think that if you put $d = \gcd(a_{1}, \ldots, a_{l})$, then you will have a solution to $$m = x_{1}^{a_{1}} \cdots x_{l}^{a_{l}}$$ if and only if there is a solution to $$m = x^{d}.$$