Different versions of $(\mathbb{Z}^{*}_{2^k}, \cdot)$

Question

I had to prove two statments, which I was supposed to use to prove two other ones:

1. For $k \geq 3, \:$ $\:5^{2^{k-3}} \equiv 1 \: \mod 2^{k-1}$, but $5^{2^{k-3}} \not\equiv 1 \mod 2^k$
2. For $k \geq 3, \:$ $\: (-1)^i5^j \equiv (-1)^m5^n \mod2^k \Rightarrow (-1)^i \equiv (-1)^m \mod4$

I have PROVED both of them, but I have to use them for the following two:

For $(\mathbb{Z}^{*}_{2^k}, \cdot)$ with $k \geq 3:$

1. ord $(\bar{5}) = 2^{k-2}$
2. $\mathbb{Z}^{*}_{2^k} = \{ (\bar{-1})^{i}\bar{5}^j | i\in\{0,1\}, 0\leq j \leq 2^{2k-2}\}=\left \langle \bar{-1}, \bar{5} \right \rangle$

I would appreciate any kind of help!