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Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
Of course. We have just defined the zeroeth power of any number as equal to 1. Since 32 = 9 and 30 = 1, we get, by ordinary arithmetic,
(32)(30) = 9 × 1 = 9.
And by our rule for multiplication of powers
(bm)(bn)
= b(m + n)(32)(30)
= 3(2 + 0) = 32.
Since 9 = 32, the results obtained by using both methods agree.
It should be clear that multiplying or dividing a number by 1 does not change the number, and that adding or subtracting 0 from the exponent will not change the number either.
Thus, our definition b0 = 1, no matter what number b stands for, works out properly in multiplication and division. (There is one exception to this. There is one value for b such that the symbol b0 is not defined at all, and that is when b = 0. The expression 00 has no meaning in arithmetic.)
Let’s try a division involving the exponent 0. What actual number do we get as an answer to the division,
(81)/(80)
using the rule for division of two powers of the same number?
