How to prove $2^{n+1} * 2^{n+1} = (2^n*2^n)+(2^n*2^n)+(2^n*2^n)+(2^n*2^n)$

Question

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 $2^{n+1} * 2^{n+1}$ = $(2^n*2^n)+(2^n*2^n)+(2^n*2^n)+(2^n*2^n)$ ?

Answer 1

We have $$2^{n+1}2^{n+1}=(2^n+2^n)(2^n+2^n)=(2^n\times2^n)+(2^n\times2^n)+(2^n\times2^n)+(2^n\times2^n)$$

Answer 2

$$ (2^{n+1})*(2^{n+1})=(2*2^n)*(2*2^n)=(2*2)*(2^n*2^n)=4*(2^n*2^n) $$

Answer 3

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 $$ 4\cdot2^{2n}=2^2\cdot2^{2n}=2^{2n+2} $$