Q1.
Why can’t n be 0?
Answer
Because the quotient rule would then ask us to divide by 0, which has no meaning.
na ÷ nb = na–b requires dividing by nb
If n = 0, then nb = 0
and division by 0 is not defined
If n = 0, then nb = 0
and division by 0 is not defined
The clearest case is a = b. Then the rule claims 0a ÷ 0a = 00, that is 0 ÷ 0 = 00. But 0 ÷ 0 has no single value — any number multiplied by 0 gives 0, so there is nothing to choose. Hence 00 is left undefined and the rules are stated with n ≠ 0.
Why it happens: notice what is being protected. The rules of exponents are shorthand for cancelling equal factors, and cancelling only works when the factor is not zero — exactly the same reason you may cancel the 3s in (3 × 5)/(3 × 7) but may not cancel the 0s in (0 × 5)/(0 × 7). Every "n ≠ 0" in this chapter is that one restriction, showing up again and again.