You are correct. (81)/(80) = 8(1 − 0) = 81 = 8.
Now that we have added zero to the list of exponents we understand, we can sum up what has been said so far:
b0
= 1 ( if b is not equal to 0)
b1
= b
b2
= b × b
b3
= b × b × b
bn
= b × b × ⋯ × b (n times, if n is a positive whole number,
or, as we say a positive integer)
and we have our two rules, one for multiplication of two powers of the same number, and one for division:
(bm)(bn)
= b(m + n)
(bm)/(bn)
= b(m − n).
So we have come to the end of Lesson 2 on the Arithmetic of Powers. In this lesson we learned to multiply powers of numbers by adding and to divide powers of numbers by subtracting.
When we raise a number to a power, we use it as a factor a certain number of times, as required by its exponent:
22
= 2 × 2 = 4
and 23
= 2 × 2 × 2 = 8
To multiply numbers written as powers of the same base, we add their exponents together to determine the total number of times the base is used as a factor in the product. Thus
(22)(23) = 25 = 2 × 2 × 2 × 2 × 2 = 32.
The formula for multiplying powers of numbers is:
(bm)(bn) = b(m + n).
Just as we added exponents in order to multiply, we subtracted exponents in order to divide. The operation of dividing
25 by
23 might be written like this:
(25)/(23) = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2) = 22.
This gives us a quotient of 2 to the second power (22). In other words, if we subtract the exponent of the divisor from the exponent of the dividend, we know to which power the base of our quotient is raised.
It is useful to state this principle in general terms so that we can apply it to any situation. The formula is:
(bm)/(bn) = b(m − n).
Sometimes when we multiply or divide powers of numbers, we obtain an exponent, or a power, of zero. This happens when we divide a power of a base by the same power of the same base, which is equivalent to dividing a number by itself. Any number divided by itself is equal to 1. Thus any number to the power zero is also equal to 1.
We can illustrate this by dividing
32 by itself first by arithmetic and then by our rule for division of powers:
(32)/(32)
= (9)/(9) = 1;
(32)/(32)
= 3(2 − 2) = 30.
Since both 30 and 1 are equal to (32)/(32), they are equal to each other. 30 = 1.
There is nothing complicated about dealing with a zero exponent when you do arithmetic with powers. To multiply together two identical bases, one of which is raised to a power of zero, simply add the exponents, thus:
(34)(30) = 3(4 + 0) = 34.
In other words, the value of 34 does not change when we multiply it by 30 just as it does not change when we multiply it by 1.
Similarly, to divide, as in the example
(bm)/(b0) = b(m − 0) = bm,
simply subtract the exponent of the denominator, which in this instance is zero, from the exponent of the numerator, with the resultant answer
bm. The value of a power is not changed by dividing by
b0, since
b0 equals 1.
Now try the Self-Test Questions for Lesson 2, in the Appendix.
