Hello I am studying cardinal arithmetic, and found out that: $$\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $$
However I found this equivalence in Wikipedia:
$$\mathfrak{c}^{\aleph_0} = \aleph_0^{\aleph_0} = n^{\aleph_0} = \mathfrak{c}^{n} = \aleph_0 \mathfrak{c} = n\mathfrak{c} = \mathfrak{c}$$
where $n$ is any finite cardinal $\geq 2$.
I do not know how they got each line of those equivalences. Could somebody help, please?
Thanks
The approach used by tetori in the comments works fine:
$$\mathfrak c \le \mathfrak c^n \le \mathfrak c^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0} = \mathfrak c$$
$$\mathfrak c \le n \mathfrak c \le \aleph_0 \mathfrak c \le \mathfrak c^2 = \mathfrak c$$