125 to the −1/3 power. You take the 1/3 power of 125, which is 5, and then take the reciprocal
of that, which is 1/5. Mathematically, it goes like this:
125 −1/3= 125 −(1/3)
= 1/(1251/3)
= 1/5
Even roots of negative numbers
What happens when you take an even root of a negative number? The simplest example of this
sort of problem is the square root of −1, but there are plenty of others. What can you multiply
by itself to get −1? Nothing that we’ve defined yet! What is the 1/4 power of 16? Again, noth-
ing we know of so far.
Mathematicians have defined quantities like this. We will explore them in Chap. 21.
They’re called imaginary numbers. They have some fascinating properties. Unlike division by
0 or the 0th root of 0, even roots of negative numbers can be “tamed.” They are commonly
used in science and engineering.
Here’s a challenge!
State the rule for negative reciprocal powers in general terms, where a is the base (the number to be raised
to the power) and p is a positive integer.
Solution
The power to which we want to raise the base is −1/p, where p is some positive integer. (We know that
−1/p will be negative, because a negative divided by a positive always gives us a negative.) If we use the
method from the above example where we evaluated 125−1/3, then we have
a−1/p=a− (1/p)= 1/(a1/p)
Multiplying and Dividing with Exponents
When we have a certain base number raised to two different powers, we get two different
quantities. But the fact that those quantities have the same base lets us take shortcuts in mul-
tiplication and division.
Multiply by adding
Let’s state the general case first, and then check out an example. Imagine a number and call it a.
This can be any number except 0. It doesn’t have to be an integer or even a rational number.
Now imagine the quantities am and an, where m and n are integers. If you multiply these two
quantities, you get the same result as if you add m to n, and then raise the base a to that power.
We can write this as an equation:
aman=a(m+n)
Multiplying and Dividing with Exponents 117
