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Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
You are correct. Our rule, (bm)/(bn) = b(m − n), gives us
(43)/(43) = 4(3 − 3) = 40
and ordinary arithmetic gives us
(43)/(43) = (64)/(64) = 1
so it appears that we should define 40 as being equal to 1. We see that 40 does have a meaning. 40 is read “4 to the power zero.”
Now, no matter what the base number b is, any power of b divided by that same power of b will, by our rule, be equal to b0, and will also be equal to 1, since it is the result of dividing a number by itself:
(b1)/(b1)
= b(1 − 1) = b0 ( by the rule);
(b1)/(b1)
= (b)/(b) = 1 ( by ordinary arithmetic)
(b3)/(b3)
= b(3 − 3) = b0 ( by the rule);
(b3)/(b3)
= (b × b × b)/(b × b × b) = 1 ( by ordinary arithmetic)
So we have reached an understanding of the 0 exponent . Any number to the power 0 equals 1. We can write
b0 = 1.
Let’s test to see that this doesn’t get us into any trouble. let’s try a multiplication, by the rule (bm)(bn) = b(m + n), where one of our exponents is 0. What result will we get by multiplying 32 by 30?
