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Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
You are correct. When we solve the problem 43 divided by 43 by ordinary arithmetic, we get the number 1 as a result:
(43)/(43) = (64)/(64) = 1.
We have already seen that when we solve the same problem by our rule for division of powers of numbers, we get 40 as a result:
(43)/(43) = 4(3 − 3) = 40.
You remember that we previously discovered that we obtained equal answers whether we worked a division problem by using the rule for division of powers or by using ordinary arithmetic; that is, when we divided 43 by 41 using the rule for division of powers, we got 42; when we divided 43 by 41 by ordinary arithmetic we got 16. And the answers 42 and 16 are equal.
Now to get back to our present problem, the division of 43 by itself, we find that we get the answers 40 and 1 as a result of our two methods of working this problem. What, then, can we say about 40?
