By defining n^0 = 1 whenever n ≠ 0, certain rules are made simpler. For example, the rule that, when you divide numbers with the same base, all you need to do is subtract the exponents, works if this rule is allowed.
For example: x^12 / x^9 = x^3, using the rule that x^a / n^b = x^(a - b). Using this rule is simpler than writing out the problem as [x·x·x·x·x·x·x·x·x·x·x·x] / [x·x·x·x·x·x·x·x·x] and cancelling out the common factors to get x·x·x, which equals x³. Do you really want to try to solve the problem x^349 / x^197 without using this rule?
Similarly, x^5 / x^8 = x^-3 by using this rule, avoiding writing out the problem in detail and cancelling out the common fractors.
Now, what about the problem x^13 / x^13. We know that the answer must be 1, because it is a number divided by itself. But, if we like to use the subtraction-of-exponents rule, we have, for an answer, x^0. Thus, x^0 must equal 1.
A different example:
10^4 = 10000
10^3 = 1000
10^2 = 100
10^1 = 10
10^0 = ??? <--- What should this be, if not 1?
10^-1 = 0.1
10^-2 = 0.01
10^-3 = 0.001
