Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Powers

Powers provide a beautiful and compact illustration of two major principles in mathematics:
  1. Introduce concepts in a simple context and then generalize them in such a way that rules and facts that are true in the simple context remain true in the more general context.
  2. Reduce your problem to one you have solved before.
In this way we obtain powerful problem solving tools, and we deepen our understanding of mathematics.

Multiplication

We start by observing that multiplication is defined by repeated addition. Thus for any number $ z $,
$\displaystyle 3\times z = z+z+z. $
More generally, for any number $ z $ and any natural number $ n $ (i.e., one of $ 1, 2, 3, \ldots $) we define the product
$\displaystyle n\times z = \underbrace{z+z+\ldots+z}_{\hbox{\( n \)~terms~\( z \)}} $
We then extend this definition to factors $ n $ that are not natural numbers (like fractions, negative numbers, or, in general, real numbers) in such a way that the ordinary rules of arithmetic remain true.

Natural Number Exponents

Powers work exactly the same way, except that instead of addition we use multiplication as our basic repeated operations. Thus we define, for any number $ z $ and any natural number $ n $,
$\displaystyle z^n = \underbrace{z\times z\times \ldots \times z}_{\hbox{\( n \)~factors~\( z \)}} $
For example,
\begin{displaymath} \begin{array}{rcccl} 2^3 &=& 2 \times 2 \times 2 &=& 8 \\ ... ... \times \ldots \times 0 & = & 0 \\ z^1 &=& z\\ \end{array} \end{displaymath}
The expression $ z^n $ is called a power and described in words as $ z $ to the power $ n $. The number $ z $ is the base of the power, and the number $ n $ is its exponent . Just like anybody should know the multiplication table it is useful to know some basic powers by heart, like those given in the following table:
\begin{displaymath} \begin{array}{rrrrrrrrrrr} n: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & ... ... 4 & 8 & 16 & 32 & 64 & 128 & 256 &512 & 1024 \\ \end{array} \end{displaymath}
Note that our definition of a power only applies when the exponent is a natural number. The base, however, can be any real number. Our task now is to extend this definition to exponents that are more general than natural numbers. We want to do this in a useful way, which means that all the rules that apply for natural number exponents also apply for more general exponents.

The Central Rule

So what are those rules? Consider, for example the product
$\displaystyle 2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 ) = 2^{3+4} = 2^ 7. $
In general it is true that (for any real number $ z $ and natural numbers $ m $ and $ n $):
$\displaystyle z^m \times z^n = z^{m+n}. $
We'll call this fact our central rule. It follows straight from the definition: we write down $ m $ factors $ z $, and then $ n $ factors $ z $, and so we write down a total of $ m+n $ factors $ z $.

Zero Exponents

Now consider the case where the exponent is zero. Read this paragraph carefully. It is elementary and understandable, and yet it illustrates a wide swath of mathematical thinking. We want our central rule to apply. For example we want to have
$\displaystyle 2^0 \times 2^3 = 2^{0+3} = 2^3. $
So $ 2^0 $ is a number that gives $ 2^3 $ when multiplied with $ 2^3 $. There is only one such number, namely the number $ 1 $. So we are compelled to define $ 2^0 = 1 $. If a base was a number other than $ 2 $, say $ z $, the same sort of argument applies, and so we define $ z^0=1 $. There is one caveat, however. If $ z=0 $ this definition would give $ 0^0=1 $. We also saw that for any natural number $ n $ we have that $ 0^n = 0. $. If we were to expand that rule to $ n=0 $ we would have $ 0^0 = 0 $. So we obtain different value of $ 0^0 $, depending on which rule we want to apply. In this case, neither rule is better than the other, and so the most useful approach is to resign ourselves to the fact that $ 0^0 $ is undefined. There is no problem when the base is non-zero however, and so we define

$\displaystyle z\neq 0 \qquad \Longrightarrow z^0 = 1. $

Rational Exponents

Now suppose the exponent is a fraction of the form $ \frac{1}{n} $. Again, our central rule applies. We have, for example,
$\displaystyle 9^{\frac{1}{2}} \times 9^{\frac{1}{2}} = 9^{\frac{1}{2}+\frac{1}{2}} = 9^1 = 3. $
So $ 9^{\frac{1}{2}} $ must be a number which gives $ 9 $ when multiplied with itself. Things now get a little murky. There are two such numbers, namely $ 3 $, and $ -3 $. By convention we pick the positive one: $ 9^{\frac{1}{2}}=\sqrt{9}=3. $ This convention is used everywhere in the world, but it does lead to confusion. For example, when we solve equations we may be interested in all solutions and simply evaluating a square root may cause us to miss some. The only way to handle this effectively is to understand the issues and to be aware and alert when solving problems. Consider another example:
$\displaystyle 8^{\frac{1}{3}} \times 8^{\frac{1}{3}} \times 8^{\frac{1}{3}} = 8^{\frac{1}{3} +\frac{1}{3} +\frac{1}{3}} = 8^1 = 8. $
Here there is only one number with that property: $ 8^{\frac{1}{3} } = 2. $ In general we define
$\displaystyle z^{\frac{1}{n}} = \root n \of z $
where $ \root n \of z\ $ to the power $ n $ equals $ z $. There is no such (real) number when $ z $ is negative and $ n $ is positive, and so for the time being we consider $ z^{\frac{1}{n}} $ undefined in that case. There are two such numbers when $ z $ is positive and $ n $ is even, and in that case we pick the positive such number. Suppose now that the exponent is a rational number whose numerator is different from 1. In that case our central rule tells us immediately what to do. For example,
$\displaystyle 8^{\frac{2}{3}} = 8^{\frac{1}{3}}\times 8^{\frac{1}{3}} = \left(8^{\frac{1}{3}}\right)^2 = 2^2 = 4. $
In general, for natural numbers $ m $ and $ n $, and $ z>0 $:
$\displaystyle z^{\frac{m}{n}} = z^{m\times\frac{1}{n}}=\left(z^{\frac{1}{n}}\right)^m=\left(\root n \of z\right)^m. $
For example,
\begin{displaymath} \begin{array}{rcccccl} 125^{\frac{2}{3}} &=& \left(\root 3 \... ...\root 5 \of 100000 \right)^3 &=& 10^3 &=& 1000\\ \end{array} \end{displaymath}

Negative Exponents

Next let's ask what a power should be if the exponent is negative. According to our central rule we should have, for example,
$\displaystyle 2^{-3} \times 2^3 = 2^{-3+3} = 2^0 = 1. $
So our unknown number $ 2^{-3} $ gives$ 1 $ when multiplied with $ 2^3 $. We have no choice but define $ 2^{-3} = \frac{1}{2^3} $. In general we define, for $ z>0 $ and any rational exponent $ r $:
$\displaystyle z^{-r} = \frac{1}{z^r}. $
For example,
$\displaystyle 8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}} = \frac{1}{4}. $
So the next time you go to a party, mention casually that you finally understand that there is just no way that $ 8^{-\frac{2}{3}} $ could equal anything other than positive one fourth. You will be the center of the conversation, and you might steer more students towards our Math 1010 classes!

Irrational Exponents

Defining powers for irrational exponents is beyond the scope of this class. However, the basic idea is to obtain a sequence of rational numbers that approaches the irrational exponent, to compute the corresponding powers, and to define the power to be that number that is approached by the powers with rational exponents.

Rules and Regulations

Intermediate Algebra Textbooks usually list tables of rules satisfied by powers. For example, it is true that
$\displaystyle \left(z^p\right)^q = z^{pq} \qquad\hbox{and}\qquad \frac{z^p}{z^q} = z^{p-q}. $
It does not pay to memorize those rules. They all flow from the original definition and the central rule. If you are unsure whether or not you can do a certain thing just ask yourself if it works when the exponents are natural numbers and you use the original definition of powers by repeated multiplication. If you think in those terms, and you do a few exercises, you will quickly learn how to work with powers confidently and effectively.