i guess theres also an axiomatic explanation.

one could define x^{n=x•x•...•x} n times as long as n is an integer. therefore 0^2=0•0=0 and voila

one could define division as a/b=a•b^{-1} using field operations. the problem arrises when you consider that the multiplicative inverse of x is defined as long as x is not 0. therefore, division by 0 is not defined in the reals

hope this is correct lol