2^0 isn't multiplying by zero. Considering this law: 2^a / 2^b = 2^(a-b)
it's obvious why 2^0 = 1
If a=b you're dividing by the same number resulting in 1.

Unfortunately, I cannot explain/prove the first law though.

[–] zalgotext@sh.itjust.works 3 points 11 months ago* (last edited 11 months ago)

The first rule is just simple division:

(2x2x2x2) / (2x2) =

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

1 * 1 * 2 * 2 =

2 * 2 =

Writing in terms of powers:

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

(2^(4-2)) =

(2^2) =

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.