I found question, that is primary question for my problem. Can't ask my question via comment to the second answer, because have not enough reputation. In proving of $$(1+q)(1+q^2)(1+q^4)\dots(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$$
I got till $1-\left(q^{2^{n+1}}\right)^2$ and thought that it equals to $1-\left(q^{2^{2(n+1)}}\right)=1-\left(q^{4^{n+1}}\right)$, but it's wrong for sure!
Now I want to understand, that I'm right about $(q^{2^{n+1}})^2=(q^{2n}q)^2=q^{2n}q^2=(q^{2^{n+2}})$
If it is possible, please explain me why is it so.
Letting $m=2^{n+1}$ then you are correct that $m^2=4^{n+1}.$
But the expression is not $q^{(m^2)}$, the expression is $(q^m)^2$, and these two expressions are not equal. In particular, we have that $(q^m)^k=q^{mk}.$
When $k=2$ we have that $(q^m)^2=q^{2m}$ and $2m=2^{n+2}.$