110 Powers and Roots
The 0th power
By convention, anything raised to the 0th power is equal to 1. Anything except 0 itself, that is!
The quantity 0^0 is not defined. You’ll see why any nonzero quantity raised to the 0th power is
equal to 1 later in this chapter. You’ll also see why 0^0 is not defined. Here are some expressions
that are all equal to 1:40
x^0 where x≠ 0
(k+ 4)^0 where k≠− 4
(abc)^0 where a≠ 0, b≠ 0, and c≠ 0
(m/n)^0 where m≠ 0 and n≠ 0
(x^2 − 2 x+ 1)^0 where x≠ 1In every expression except the topmost one, there are constraints on the variables in the quan-
tities being raised to the 0th power. These keep the values of the quantities from being equal
to 0. If you’re not sure about the reason for the constraint on x in the last expression, hold on
a minute and you’ll see.
Whenever you find any expression containing variables, and that expression is to be raised
to the 0th power, be sure you never let that expression attain a value of 0. This is especially
important if such an oversight is a step in solving a problem! You would be throwing an unde-
fined quantity into a sensitive process. You’ve heard what computer programmers say about
putting nonsense into a machine! The same thing happens in mathematics.Negative integer powers
If a is any number and n is a negative integer, the expression an means the reciprocal of the
quantitya raised to the power of |n|. For example,a−^1 = 1/(a^1 )= 1/a
a−^3 = 1/(a^3 )
a−^20 = 1/(a^20 )As before, a can be almost any expression you can imagine. Note that the −1st power of any
quantity is the same thing as its reciprocal (multiplicative inverse). You’ll often see this nota-
tion used because it can be a lot less “messy” than writing 1, then a slash, and then a compli-
cated expression. Here are some examples of numbers or quantities raised to negative integer
powers:4 −^2
x−^4 where x≠ 0
(k+ 4)−^7 where k≠− 4
(abc)−^4 where a≠ 0, b≠ 0, and c≠ 0
(m/n)−^12 where m≠ 0 and n≠ 0
(x^2 − 2 x+ 1)−^5 where x≠ 1