Win At Mathematics

Here is the solution: But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this for me please?

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you

Let $W \leq \mathbb{R}^n$ and set $$W^{\perp} = \{v \in \mathbb{R}^n | v.w =0, \hspace{0.25cm} \forall w \in W\}$$ (a) Show that $W^{\perp} \leq \mathbb{R}^n$ . (b) Let $v_1=\langle 1, 1, 1, 2\rangle$ , $v_2=\langle 2, 1, 1, 1\rangle$ , and $v_3=\langle -1, 2, 2, 7\rangle$ . Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^4$ be the transformation given by $e_1 \mapsto v_1$ , $e_2 \mapsto v_2$ , and $e_3 \mapsto v_3$ .Let $W=img(T)$ . Find a basis for $W$ and $W^{\perp}$ .

Let $W \leq \mathbb{R}^n$ and set $$W^{\perp} = \{v \in \mathbb{R}^n | v.w =0, \hspace{0.25cm} \forall w \in W\}$$ (a) Show that $W^{\perp} \leq \mathbb{R}^n$ . (b) Let $v_1=\langle 1, 1, 1, 2\rangle$ , $v_2=\langle 2, 1, 1, 1\rangle$ , and $v_3=\langle -1, 2, 2, 7\rangle$ . Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^4$ be the transformation given by $e_1 \mapsto v_1$ , $e_2 \mapsto v_2$ , and $e_3 \mapsto v_3$ .Let $W=img(T)$ . Find a basis for $W$ and $W^{\perp}$ .