Exponentiation: what is $2^{2^{k+1}}$, $2^{2^{k+2}}$, and $2^{2^{k + n}}$?

Question

$2^{2^{k+1}}=({2^{2^k}})^2$

$2^{2^{k+2}} = ?$

$2^{2^{k+n}} = ?$

Just can't figure it out.

Answer 1

Hint. Recall that $a^{b+c}=a^b\cdot a^c$ and $a^{b\cdot c}=(a^b)^c=(a^c)^b$.

Hence we have $2^{2^{k+n}}=2^{2^{k}\cdot 2^n}$. Can you take it from here?