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Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
You are correct. By following the rule (bm)/(bn) = b(m − n) we get
(b3)/(b2) = b(3 − 2) = b1.
To see why this is true let’s write out the powers completely:
(b3)/(b2) = (b × b × b)/(b × b)
← dividend
← divisor
Each time b occurs in the divisor, it cancels a b in the dividend.
(b3)/(b2) = (b × b × b)/(b × b) = b1 ← quotient
So the number of times b occurs as a factor in the quotient is simply the number of b’s left after the cancellation.
We don’t have to do this blindly, of course. We can check by ordinary arithmetic. Let’s try dividing 8 = 23by 4 = 22. The quotient will be (8)/(4) = 2, and as you can see:
(8)/(4) = (23)/(22) = 2(3 − 2) = 21 = 2.
It works out, doesn’t it? We get the same answer whether we divide by ordinary arithmetic, or by using exponents.
What regular number do you get by dividing 128, which is 27, by 16, which is 24? Do the problem by means of exponents.
