Why should the answer be 0? We have an intuition that 0 is the identity element for addition, so it makes sense to treat it as the starting point for many thin

Why should the answer be 0? We have an intuition that 0 is the identity element for addition, so it makes sense to treat it as the starting point for many things. However, when we look at multiplication, 0 looks very different. Instead of preserving the other value, as it does with addition, multiplying by 0 changes the other value in an information destroying way (e.g if you know that x * 0 = 0, you have no idea what x is). This property of 0 is so distinct from the other numbers, that when we consider the integers (or reals, or complex numbers) as a structure that only has multiplication, but not addition, we typically exclude 0 from consideration. Since (discrete) exponentiation is a multiplicative process, it makes sense to consider it in terms of the integers under multiplication; and so excluding 0. When we consider integers under multiplication, we notice that 1 has the same properties that 0 had under addition. Namely, 1 * x = x for all x. Given this, it makes sense to treat 1 as the "empty" element, the same way you want to do with 0 in additive contexts.