This is a really simple question, but I don't know how to phrase it well enough for Google. I'm going through a proof and don't understand how:
$$ (q^{2^{n+1}})^2 = q^{2^{n+2}} $$
I thought it would be $q^{4^{n+1}}$instead, or are they equivalent? If so, why?
They are not equivalent because $2^{n+2}\not= 2^{2n+2}=2^{2(n+1)}=(2^2)^{n+1}=4^{n+1}$.
Use the followings carefully :
$$(q^a)^b=q^{a\color{red}{\times} b}$$ and $$2^c\times 2^d=2^{c\color{red}{+}d}.$$
$$\big (q^{2^{n+1}})^2=q^{2^{n+1}\times 2}=q^{2^{n+1}\times 2^1}=q^{2^{n+1+1}}=q^{2^{n+2}}$$
They are not equivalent.
Nested powers work like this:
$$(a^b)^c = a^{bc}$$
Note that this is different from
$$a^{(b^c)} = a^{b^c}$$
In your example,
\begin{align*} \left(q^{2^{n+1}}\right)^2 &= q^{2 \cdot 2^{n+1}}\\ &=q^{2^{n+2}} \end{align*}
To answer your question, we can look at an example.
Let $q=3$, and $n=1$. Then $$q^{2^{n+1}} = 3^{2^{2}} = 3^4 = 81.$$
If we square this we get $$\left(q^{2^{n+1}}\right)^2 = (3^{4})^2 = 81^2= 6561.$$
However, notice that $$q^{4^{n+1}} = 3^{4^2} = 3^{16} = 43046721.$$
This is a bit larger than $6561$, so the two expressions are not equal.
