You’re guessing. Think how we became tangled up with this 40 in the first place. We had a rule for dividing one power of a number by another power of that same number:
(bm)/(bn) = b(m − n)
This rule worked fine so long as m was larger than n. For example, we had
(43)/(41) = 4(3 − 1) = 42
which is correct, as we can find out by checking by ordinary arithmetic:
(43)/(41) = (4 × 4 × 4)/(4) = (64)/(4) = 16
and 16 is equal to 42, isn’t it?
Now, when we came to a case where m was equal to n, such as 43 divided by itself, we got
(43)/(43) = 4(3 − 3) = 40.
We have not defined 40 yet, but we reached it by dividing a power of 4 by itself. However, any number divided by itself equals 1, doesn’t it?
(43)/(43) = (64)/(64) = 1.
So (43)/(43)is equal to 40 by our rule, and is equal to 1 by ordinary arithmetic. Now, that tells us how to define 40, doesn’t it?
