Explained: The "Laws" of Exponents

If you've ever taken an algebra course, you certainly encountered the so-called Laws of Exponents. A list of laws or rules which dictate how various mathematical expressions can be manipulated, simplified, or rewritten. Such lists usually look like

  1. $x^m x^n = x^{m+n}$
  2. $(x^m)^n = x^{mn}$
  3. $x^0 = 1$
  4. $x^{-m} = \frac{1}{x^m}$
  5. $\frac{x^m}{x^n} = x^{m-n}$
  6. $x^{m/n} = \sqrt[n]{x^m}$

For example, these laws can be used to rewrite expressions like $4^3 \cdot 4^7$ as $4^{10}$ or $(x^5)^{10}$ as $x^{50}$. These laws are usually presented as a list with an accompanying example illustrating how each law is used but an explanation as to why these laws exist at all is not given.

In the title, I used quotations around the word 'laws' because I hate the word law when it's used in mathematics. Mathematics has no laws. Mathematics has consequences of definitions. I'll illustrate this by examining and explaining why the "laws" of exponents are what they are. It's all because of our definition of what an exponent (or power) means.

Let's say we want to multiply the number 2 by itself 10 times. That's excessive but it's manageable

$$ 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2 $$

But what if we needed to multiply 2 by itself 100 times? Again, we know how to do it, it's just very excessive

$$ \underbrace{2\cdot2\cdot2\dots2}_\text{100 copies of 2} $$

Yuck. It's not that we can't perform the arithmetic, but writing this instruction "multiply 2 by itself 100 times" is too cumbersome. Mathematicians have and always will be the good kind of lazy. They want to convey as much information as possible using the least amount of "stuff" (ink, letters, space, etc). So mathematicians do what they do and they create abbreviations for the cumbersome, making it less so.

Thus an abbreviation for "multiply 2 by itself 100 times" is born. It was chosen to be $2^{100}$. This is simply an abbreviation for the sentence "multiply 2 by itself 100 times". In general the expression $x^n$ means "multiply $x$ by itself $n$ times."

What's implicit in the above abbreviation $x^n$ is that $x$ represents anything and, for the time being, it only make sense if $n$ is a positive whole number. E.g. $2^4$ makes sense, it's just 2 multiplied by itself 4 times. But what the does $2^{-3.4}$ mean? 2 multiplied by itself -3.4 times? We'll get there. For now, $x$ is absolutely anything and let's restrict $n$ only to be a positive whole number.

What we have just done is defined the following abbreviation:

$$ x^n = \underbrace{xxx\dots x}_\text{$n$ copies of $x$} $$

Let's ask and answer some questions which will generate the entire list of the laws of exponents. The only thing we need to know is the above definition of our abbreviation for $x^n$.

If $m$ and $n$ are positive whole numbers, what is the meaning of $x^m x^n$? Well,

x^m x^n &= \underbrace{\underbrace{xxx\dots x}_\text{$m$ copies of $x$}\cdot\underbrace{xxx\dots x}_\text{$n$ copies of $x$}}_\text{$m+n$ copies of $x$} \\
&= x^{m+n}

So the first law is just a consequence of what we defined $x^n$ to mean.

Again, assuming $m,n$ are both positive whole numbers, what is the meaning $(x^m)^n$? Well,

(x^m)^n &= (\underbrace{xxx\dots x}_\text{$m$ copies of $x$})^n \\
&= \underbrace{(\underbrace{xxx\dots x}_\text{$m$ copies of $x$})\cdot (\underbrace{xxx\dots x}_\text{$m$ copies of $x$})\cdot (\underbrace{xxx\dots x}_\text{$m$ copies of $x$}) \dots (\underbrace{xxx\dots x}_\text{$m$ copies of $x$})}_\text{$mn$ copies of $x$} \\
&= x^{mn}

So the second law is just a consequence of what we defined $x^n$ to mean. You may be sensing a pattern.

Now for the fun part: why anything raised to the zeroth power is equal to one, or equivalently, why $x^0 = 1$. We'll accomplish this indirectly by examining the meaning of $x^0 x^n$, again assuming $n$ is a positive whole number.

x^0 x^n &= x^{0+n} && \text{first law of exponents} \\
x^0 x^n = x^n \\
x^0 x^n - x^n &= 0 \\
x^n(x^0 - 1) &= 0

The last line says a product of two things is equal to zero. This happens only when either thing is itself equal to zero, i.e. either $x^n = 0$ (and this happens only when $x=0$) or if $x^0 = 1$. We want our conclusion to be valid for any value of $x$. This means $x^0 = 1$ for every possible value of $x$. In particular this also shows that $0^0 = 1$.

What of negative exponents? Let's assume $n$ is a positive whole number. We'll indirectly investigate the meaning of $x^{-n}$ by examining the meaning of $x^n x^{-n}$.

x^n x^{-n} &= x^{n-n} && \text{first law of exponents} \\
x^n x^{-n} &= x^0 \\
x^n x^{-n} &= 1 && \text{third law of exponents}\\
x^{-n} &= \frac{1}{x^n}

This is yet another consequence of our definition of the abbreviation for $x^n$. Additionally, this consequence removes our current restriction of exponents being positive whole numbers, they can now be any whole number, positive, negative, or zero.

Now, what of $\frac{x^m}{x^n}$? Assuming $m,n$ are whole numbers, we have

\frac{x^m}{x^n} &= x^m \frac{1}{x^n} \\
&= x^m x^{-n} && \text{fourth law of exponents} \\
&= x^{m+(-n)} && \text{first law of exponents} \\
&= x^{m-n}

Again, this is just a consequence of the definition for our abbreviation of $x^n$.

Lastly, what of fractional exponents? Let's assume $m,n$ are whole numbers, with $n\neq 0$. Again, the meaning of $x^{m/n}$ will be investigated indirectly by examining the meaning of $(x^{m/n})^n$:

(x^{m/n})^n &= x^{mn/n} && \text{second law of exponents} \\
&= x^m

The last lines is asserting that $x^{m/n}$ when multiplied by itself $n$ times yields $x^m$. This is another way of saying that $n$th root of $x^m$ is $x^{m/n}$, or equivalently, $x^{m/n} = \sqrt[n]{x^m}$.

Whew. That was a lot. It's not hard to explain or to see why these "laws" are the result. It's also understandable that such an in depth investigation into why these "laws" of exponents are what they are isn't usually afforded during class time. It's my hope these explanations will help those who struggled to understand some of the less intuitive results, namely laws three through six.

This list of six "laws" of exponents help explain what it means to raise a number $x$ to any number $y$ except for one class of numbers (or two classes, if we consider complex numbers).

There are numbers in existence that cannot be expressed as the ratio of two whole numbers, i.e. a fraction. These numbers are called irrational. Familiar examples of irrational numbers are $\sqrt{2}$ and $\pi$. The exact meaning of $x^\sqrt{2}$ or $x^\pi$ is outside the scope of this blog post. But will be explained in greater detail in a future blog post, check back later!