Show that $ a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $ Show that $  a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $
Show that$  a \equiv 1 \pmod{2^4 } \Rightarrow a^{2^{4-2}} \equiv 1 \pmod{2^4} $
Answer:
$ a \equiv 1 \pmod{2^3} \\ \Rightarrow  a^2 \equiv 1 \pmod{2^3} \\ \Rightarrow a^{2^{3-2}}=a^{2^1} \equiv 1 \pmod{2^3} $
Am I right?

Yes, it is correct. 
In general, if $a \equiv 1 \pmod m$,then we have $a^n \equiv 1 \pmod m$ for any $n$.
