Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
Page 86
Well, first, this is a problem in division. Our rule for dividing one power of a number by another power of the same number is
(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 tried the case in which
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, 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. And that tells us how to define 40, doesn’t it?
Now return to Page 80 and try again.
