By using this site, you are accepting "session" cookies, as set out in the site's Privacy Policy
Cookies are also used to remember which page of the book you last viewed, so that when you revisit the site you automatically return to the last page you visited.
Close
Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
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?
