Edit1: Used Latex. =] Edit2: Thanks for the guidance to the users below. Really helped me out editing the post and guidance on the math problem.
The question gave me a hint: $a^{2^{n+1}} = (a^{2^n}) ^ 2 $
This one is a lot harder than the ones given in the examples...
Comments: At first I thought the hint said $a^{2^{n+1}}$ = $(a^{2^{n}})$ * $1^{2}$ but it does not seem to be the case.
My approach: I tried leveraging the hint by solving it in terms of $a^{2^{n}} $ which would mean $a^{2{^n}}$ = $a^{2^{n+1}}$ / $(1)^2$. But it does not look right.
What I know from reading the textbook: If I can find what $a^{2^n}$ represents, writing the recursive definition will be a lot easier.
Consider the sequence $a,a^2,a^{4},a^8,\cdots$ - what do we do to go from one term to the next?