Power Rules

Powers are very important in algebra. We begin by defining the power of a symbol by the count of the number of times that symbol is to be multiplied by itself, that is:

$$a^n = a*a*...*a*a \quad (n {\rm\ times})$$ (3.29)

where $n$ (the power) is obviously some integer, since we don't really know what it would mean (yet) to multiply out anything but discrete numbers.

However, the rules for multiplication also include division as multiplication of inverses. We've already indicated that the inverse relationship can be written as a negative power, $1/a = a^{-1}$. From this we can see that:

$$a^{-n} = \frac{1}{a*a*...*a*a} \quad (n {\rm\ times})$$ (3.30)

makes sense, again for $n$ any integer.

Now consider the following expression:

$$b = a^n = a*a*...*a*a \quad (n {\rm\ times})$$ (3.31)

Note first of all that if $a$ is any (say real) number, $b$ will definitely be a real number because the reals are closed under multiplication. It is less obvious, but still true, that if $b \ge 0$ there is always at least one number $a > 0$ such that this equation is satisfied (you'll have to take my word for it for now, although later on you'll come to understand this in more detail if you ever need to).

If $b < 0$, we have a bit of a problem. If $n$ is odd, once again there will always be at least one $a < 0$ for which this equation is true because the product of an odd number of negative numbers is negative. If $n$ is even and the real number $b$ is negative, however, we are in trouble. That is because there is no real number $a$ such that e.g.

$$-1 = a^2$$ (3.32)

Squaring a real number produces a non-negative real number as a result!

This, of course, is why we invented complex numbers. They are extended from the reals so they contain numbers like the imaginary unit $i$ defined by:

$$-1 = i^2$$ (3.33)

With or without this complex extension, we'll find it very useful to define the following $a$ that ``solves'' $b = a^n$:

$$a = b^{1/n}$$ (3.34)

A real solution $a$ will always exist for $n$ odd and any real $b$. A real solution $a$ will always exist for $n$ even and a real $b \ge 0$. No real solution $a$ will exist for $n$ even and real $b < 0$, which is then a restriction on the domain of fractional powers over the set of all real numbers. Finally, a solution will always exist for any $b$ and any $n$ if $a$ (and/or $b$) are permitted to be complex.

With this definition of fractional powers, we can now define the following algebraic rules that work for powers. These rules are very important and extremely useful. We defer any discussion of allowed domains or real versus complex until the end. We begin with:

$$0^n = 0 \quad\quad(n>0)$$ (3.35)

$$0^n = \frac{1}{0} = {\rm undefined} \quad\quad(n \le 0)$$ (3.36)

There exists a sense where the latter can stand for $\infty$, but it is by far the safest to just say it is undefined, since $\infty$ in the sciences is generally not a number although (sigh) mathematicians have lots of harmless fun pretending that it is. In fact, we'll talk about $\infty$ later on in this work so you really understand the way the concept should be used in almost all practical applications so that its use yields an unambiguous answer. Ultimately this means that powers of zero are zero, the inverse of (or division by) zero is meaningless.

With a ``special'' pair of rules for $a = 0$ in hand above, we will now insist that $a \ne 0$ in all the rules below and that (at first) $m$ and $n$ be integers:

$$a^0 = 1$$ (3.37)

$$a^m a^n = a^{(m+n)}$$ (3.38)

$$a^m a^{-n} = a^{(m-n)}$$ (3.39)

$$(a^m)^n = a^{mn}$$ (3.40)

$$(a^{\frac{1}{m}})^n = a^{\frac{n}{m}}$$ (3.41)

This last rule is extremely interesting to us. It tells us that subject to the domain restrictions above that allow us to define $a^{1/n}$ in the first place, $a^r$ is a meaningful number for any rational number $r$! Let's summarize the domain rules one more time as they are critically important:

1. If real number $a > 0$ there is always a real number $b = a^r$.
2. If real numb er $a < 0$ and if the rational number $r$ has an odd denominator or is an integer, there is always a real number $b = a^r$.
3. If $a \ne 0$ is complex, there is always a complex $b = a^r$ for any rational $r$.

Don't (please!) try to memorize these rules. Just study them until you understand them and can remember them as a part of that understanding.

All of the algebraic power rules listed above therefore hold for any rational numbers $m$ and $n$ and not just integers, subject to these domain rules. This leaves us with an open question. We have proven that for very general domain restrictions (usually we just say $a > 0$ and forget about all the rest) $a^r$ exists for for any rational number $r$. What about $a^x$ for any real number $x$?

From a computational point of view it hardly matters. Any number you can punch into a calculator or program into a computer is clearly OK, and one can get (in some sense) ``as close as you like'' to any irrational number. So to a physicist or scientist, we just plain don't care. Mathematicians are a bit pickier - does (say) $a^\pi$ exist? Note that in this case there is no denominator so that we can write this out as an integer power of the inverse of an integer power of a number, where all we originally defined the notion of the ``power'' of a number as being a certain integer number of products of that number. Our original notion of power has morphed into something quite new.

Without proof, the answer is pretty much yes. As is not infrequently the case the ``best'' way to think of powers is to define powers by the rules above on the most general complex domain, then consider special cases of this definition that work for e.g. real $a > 0$.

In fact, we'll do one more extension and a bit of simplification as we no longer need to worry showing that rational exponents work. Let us note that $b = a^x$ where $a$, $b$ and $x$ are complex numbers (yes, the exponent too!) always has one or more solutions. In that case:

$$a^0 = 1$$ (3.42)

$$a^x a^y = a^{(x+y)}$$ (3.43)

$$(a^x)^y = a^{xy}$$ (3.44)

holds on the complex plane. We'll have to wait until later to define just what is meant by taking a number to the $i$th power and the inverse relation implied by these rules, our friend the logarithm.

In the meantime, let's see what transformations these rules enable us to perform on equations containing powers.