The AOE and SOE rules work not only when the exponents are integers, but for any rational numbers.
You might call these facts the generalized addition-of-exponents (GAOE) rule and the generalized subtraction-
of-exponents (GSOE) rule.
Here’s a challenge!
Using the SOE rule, provide a demonstration of why any nonzero quantity to the 0th power is equal to 1.
Also show why 0^0 is not defined.
Solution
Look again at the formula that “translates” division of quantities into subtraction of exponents. That
formula is
bp/bq=b(p−q)
where b is the base and p and q are integers. Now let’s think of the formula in reverse. We can transpose
the left-hand and right-hand sides of the equation to get
b(p−q)=bp/bq
There’s nothing in the “rule book” that says we can’t have p and q be the same. Let’s do that, and call them
bothp. Then we have
b(p−p)=bp/bp
The left-hand side of this equation is b raised to the (p−p)th power, which must be b raised to the 0th
power because p−p is always 0. The right-hand side is bp divided by itself, which has to equal 1 as long
asb≠ 0.
Now we get to the 0^0 situation. Let’s violate the “rule book” and let b= 0 in the above equation. Then
we get
0 (p−p)= 0 p/0p
No matter what nonzero value we choose for p, we get 0^0 on the left-hand side of this equation, and 0/0
on the right. So
00 = 0/0
The quantity 0/0 is not defined, so 0^0 can’t be, either.
Multiple Powers
Numbers can be raised to powers more than once. In this section we’ll see what happens when
you raise a quantity to a power, and then raise the result to another power.
Multiple Powers 119
