Question about the identity of $2^{2^n}$

Question

Is this identity true: $2^{2^n}= 4^n$? I believe this is true as far as I know. Sorry this is the only place to ask.

Is there another identity for $2^{2^n}$ which I can simplify to?

Answer 1

Consider $n = 0$. The equation would not work.

Answer 2

Hint: What happens when $n = 0$?

Advice:

Also, before asking simple questions like these, it is a good idea to try out a few values.

Answer 3

A compiler — i.e., a computer program that translates a program written in a high-level language into a machine-level language — treats a^b^c as a^(b^c) rather than as (a^b)^c, (as the OP speculated).

Answer 4

No. In standard usage, $2^{2^n}$ means $2^{(2^n)}$, not $(2^2)^n$.

Notice that $(a^b)^c$ is equal to $a^{bc}$, whereas $a^{(b^c)}$ cannot be similarly simplified.