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Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
If you’ll remember, we had established that 41 = 4.
Now as for 40, think how we got entangled with it in the first place. We had a rule for dividing one power of a number by another power of the 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 by checking by ordinary arithmetic:
(43)/(41) = (4 × 4 × 4)/(4) = (64)/(4) = 16
and 16 is equal to 42, isn’t it?
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 yet defined this 40, which we reached by dividing a power of 4 by itself. But 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, if (43)/(43) equals 1 and also equals 40, we have discovered how to define 40, haven’t we? Return to Page 80 and select the right answer.
