The first rule is just simple division:

(2x2x2x2) / (2x2) =

(2/2) * (2/2) * 2 * 2=

1 * 1 * 2 * 2 =

2 * 2 =

4

Writing in terms of powers:

(2^4) / (2^2) =

(2^(4-2)) =

(2^2) =

4

The two bottom 2's "cancel out" (really they just divide into one another to make 1's) two of the top 2's and you're left with two top twos.