How to prove $2^{n+1} * 2^{n+1} = (2^n*2^n)+(2^n*2^n)+(2^n*2^n)+(2^n*2^n)$ Below diagram is used as part of a proof of induction to prove that $E$ a way to tile a $2^n * 2^n$ region with square missing : 

What is the proof that $2^{n+1} * 2^{n+1}$ = $(2^n*2^n)+(2^n*2^n)+(2^n*2^n)+(2^n*2^n)$ ?

In general, $a^m\cdot a^n=a^{m+n}$, so $a^n\cdot a^n=a^{n+n}=a^{2n}$. Thus
$$
a^n\cdot a^n+a^n\cdot a^n+a^n\cdot a^n+a^n\cdot a^n=4a^n\cdot a^n=4a^{2n}
$$
Similarly, $a^{n+1}\cdot a^{n+1}=a^{2n+2}$. Now let $a=2$; the former expression is
$$
4\cdot2^{2n}=2^2\cdot2^{2n}=2^{2n+2}
$$
