If I have a function defined as ,
$U_{n} = \{(x_{1},x_{2},...,x_{n}) \in \mathbb{N}_{0}^{n} | x_{1} + 2x_{2} + 3x_{3} + ... + nx_{n} = n\}$
what is the meaning of having $\mathbb{N}_{0}^{n}$ raised to the $n$ power? I know the zero tells us that we can include zero but I don't know what the $n$ is telling us.
By way of example, $1 \in \mathbb{N}$, but $(1,1) \in \mathbb{N}^2$.
It's a shorthand for the cross product, and indicates what we might think of naturally as an $n$-dimensional number, each of whose entries is a natural number.
Similarly, $(1, 1, \pi, e, e) \in \mathbb{N}^2 \times \mathbb{R}^3$.