Algebra Know-It-ALL

The constraints are imposed, as with the 0th power, to keep the “powerized” quantities from
being equal to 0. If you take a negative integer power of 0, you end up with 1/0, and that’s not
defined. For example, 0−^7 = 1/(0^7 )= 1/0.

Are you confused?

You should be able to see, without much trouble, how the second, third, fourth, and fifth quantities above
can be equal to 0. Here they are:

x= 0 when x= 0, of course!

(k+ 4) = 0 when k=− 4

(abc)= 0 when a= 0, b= 0, or c= 0

(m/n)= 0 when m= 0

But what about the sixth and last expression? It’s not immediately clear that

(x^2 − 2 x+ 1) = 0 when x= 1

You can “plug in” the value 1 for x and see that you get 0 when you add everything up. But would
you have “plugged in” 1 at random, thinking it might cause the whole expression to equal 0? Prob-
ably not!
When you deliberately allow this whole expression to be equal to 0, you get something called a qua-
dratic equation. Such equations can be solved in various ways. It turns out that there is one solution to
the equation

(x^2 − 2 x+ 1) = 0

That happens to be x= 1. Don’t worry about how this type of equation can be solved right now. You’ll
learn how to do it later in this book. For the moment, pay attention to the important message: Beware of
taking 0 to the 0th power, and beware of dividing by 0! Don’t let these things happen, even accidentally.

Here’s a challenge!

What happens if you start with 2 and raise it to successively higher positive integer powers? What happens
if you raise 2 to integer powers that get larger and larger negatively?

Solution

If you start with 2 and raise it to positive integer powers, you get

21 = 2

22 = 2 × 2 = 4

23 = 2 × 2 × 2 = 8

24 = 2 × 2 × 2 × 2 = 16

↓

and so on, forever

Integer Powers 111