Laws of Indices for Rational Exponents

Laws of indices for all rational exponents

Indices [or powers, or exponents, if you prefer] are really a mathematician's method for writing out certain ways of combining numbers without using vast quantities of ink [mathematicians like beauty, clarity, precision, elegance, and brevity; saving ink whilst maintaining these virtues is the ideal]. They are a good example of what a well-chosen notation can do. A well-chosen notation aids thinking and makes calculations and manipulations easier than they might otherwise be. For the TMUA/ESAT you are expected to know all the basic rules of indices – both what the notation means and how to deal with the notation.

In this section, we will introduce the basic rules we expect you to know along with some informal notes to help you start to think about how the ideas fit together.

We start with the very basic idea of an index for a number $a$ multiplied by itself a total of $m$ times [that is, $a$ appears $m$ times in the expression]:

$$\frac{a \text{ appears } m \text{ times here}}{a \times a \times a \times \dots \times a}$$

We write this concisely as $a^m$.

This basic idea allows us to work out how we might combine powers when we multiply: we can ask how we might write $a^m \times a^n$. If we write the whole expression out term-by-term and then use the rule that $a \times a \times a \times \dots \times a$ $m$-times is written as $a^m$ we arrive at the following:

$$a^m \times a^n = \underbrace{(a \times \dots \times a)}_{m \text{ times}} \times \underbrace{(a \times a \times \dots \times a)}_{n \text{ times}} = \underbrace{a \times a \times a \dots \times a}_{m+n \text{ times}} = a^{m+n}$$

And so, we have our first rule for our notation – a rule that is really just the direct consequence of how we decided to write $a \times a \times a \times \dots \times a$ in our notation:

$$\text{RULE 1} \quad a^m \times a^n \equiv a^{m+n}$$

We note that, for the moment, this rule applies when $m$ and $n$ are whole positive numbers. Later we will explain that the rule works for ALL real numbers $m$ and $n$.

Next, we are going to extend this rule to 'invent/derive' and motivate some other notation. We want to be able to use this rule when $m$ and $n$ are not integers and we also want to make sure that our notation is consistent – that is, we don't want to find that we introduce definitions and rules that give us different answers depending on how we apply them.

The first thing we are going to decide is that RULE 1 works when $a$ is positive and $m$ and $n$ are ANY rational number – there are good reasons for this decision which we shall talk about later. That means, for instance, we can apply the rule when $m$ is $\frac{1}{3}$ and $n$ is $-\frac{31}{17}$ and so on. But we do need to ask, what does $a^{\frac{1}{3}}$ mean, and what does the minus sign in $a^{-\frac{31}{17}}$ mean? We will motivate our answers using RULE 1 as this will ensure that our use of notation extended to fractional powers is consistent.

Let's start by trying to work out what we would like $a^{\frac{1}{3}}$ to mean. We can use RULE 1 extended to fractions to write:

$a^{\frac{1}{3}}a^{\frac{1}{3}}a^{\frac{1}{3}} = a^{\left(\frac{1}{3} + \frac{1}{3} + \frac{1}{3}\right)} = a^1 = a$

And this means that we must interpret $a^{\frac{1}{3}}$ as the cube-root of $a$ [and note, we also used $a^1 = a$].

So, it should be "obvious" that we must interpret $a^{\frac{1}{n}}$ as the $n^{\text{th}}$ root of $a$; this gives us our second rule:

$RULE\ 2 \quad a^{\frac{1}{n}} = \sqrt[n]{a}$

Next, we are going to look at what happens when we introduce a minus sign into the power. We are going to do this by exploring how RULE 1 might fit together with expression such as $a^3 \times a^{-2}$:

Using RULE 1 extended to negative numbers we obtain the following:

$a^3 \times a^{-2} = a^{3+(-2)} = a^1 = a$

And we ask ourselves what we need to multiply $a^3$ by to get $a$; and the answer is that we need to multiply $a^3$ by $\frac{1}{a^2}$ to get an answer of $a$. This suggests that we should interpret $a^{-2}$ as being the same as $\frac{1}{a^2}$ and leads to our third rule:

$RULE\ 3 \quad a^{-m} = \frac{1}{a^m}$

Again, this only works when $a$ is a positive number.

Next, we will tackle $a^0$. To do this, we will use RULE 1 and RULE 3:

$a^2 \times a^{-2} = a^{2+(-2)} = a^0$

But we can also look at this another way:

$a ^ { 2 } \times a ^ { - 2 } = a ^ { 2 } \times { \frac { 1 } { a ^ { 2 } } } = { \frac { a ^ { 2 } } { a ^ { 2 } } } = 1$

Now recall, we must make sure that all the definitions and rules we use are consistent – that is to say, we get the same answer no matter how we tackle a question using our rules. This means we must have $a ^ { 0 } = 1$.

There are other ways of deciding or justifying that $a ^ { 0 }$ must have the value 1 and we will touch upon some of these ideas at the end of this section when we look at how we might extend the rules to cases where the powers are irrational numbers.

We now have RULE 4:

$R U L E 4 \quad a ^ { 0 } = 1$

Again, this works only when $a$ is a positive number.

You should now have enough information to understand the other rules of indices:

$R U L E 5 \quad a ^ { m } \div a ^ { n } = { \frac { a ^ { m } } { a ^ { n } } } = a ^ { m } \times a ^ { - n } = a ^ { m - n }$

$R U L E 6 \quad ( a ^ { m } ) ^ { n } = { \underbrace { a ^ { m } \times a ^ { m } \times \ldots \times a ^ { m } } _ { n \; t i m e s } } = a ^ { { \frac { n \; t i m e s } { m + m + \cdots + m } } } = a ^ { m n }$

$R U L E 7 \quad a ^ { \frac { m } { n } } = ( a ^ { m } ) ^ { \frac { 1 } { n } } = { \sqrt [ n ] { a ^ { m } } } = \left( a ^ { \frac { 1 } { n } } \right) ^ { m } = \left( { \sqrt [ n ] { a } } \right) ^ { m }$

With one of these rules [RULE 6], it is important to be a little careful as sometimes it is possible to misinterpret the notation: consider the two expressions $( a ^ { m } ) ^ { n }$ and $a ^ { m n }$. It is easy to think that these two expressions mean the same thing as they look very similar and often look almost the same when they are written out on paper. But they mean different things:

$( a ^ { 3 } ) ^ { 2 } = a ^ { 3 } \times a ^ { 3 } = a ^ { 6 } \qquad a ^ { 3 ^ { 2 } } = a ^ { ( 3 \times 3 ) } = a ^ { 9 }$

Finally, we have been careful throughout this section to emphasise that the rules only work when $a$ is a positive number [rational or irrational]. And our rules allow us to understand the meaning of $a ^ { m }$ when $m$ is any positive or negative rational numbers or zero [RULE 3 and RULE 7 are useful here]. In fact, even though the specification restricts things to rational powers only, the rules work for any positive $a$ and any real [rational or irrational] $m$ and $n$.

Because the specification says "rational exponents" we are careful in TMUA/ESAT questions to ensure that these issues [that powers are rational, but can be irrational] are not ones that you need to think about; that is to say, even though we do set questions where $m$ and $n$ could be irrational, we are careful to ensure this fact does not get in the way of your ability to answer the question.

The rest of this section is NOT part of the TMUA/ESAT specification, so you can skip it if you want. But it is useful and gives you some insights into how mathematicians think about things.

We are going to answer the question: what happens to our rules when $m$ or $n$ are not rational, and what happens when $a$ is zero or even when $a$ is a negative number? This will help you understand why we only have positive $a$ values but we let $m$ and $n$ be any real number.

The answer is that things get more complicated in some cases but not others. We will look at two cases and make some brief comments on some of the others:

Case 1

What happens when $a$ is positive but $m$ and $n$ are irrational?

The answer is [as we have already mentioned] that all the rules still apply, and we interpret $a^m$ [and $a^n$] where $m$ is irrational in a clever way. Let's look at how we might interpret $2^{\sqrt{3}}$. We will tackle this by looking at how we deal with graphs of exponential functions. You will probably have met the graph of $y = 2^x$ and sketched it but with no thought about whether $x$ is rational or whether $x$ is irrational. If we try to sketch the graph of $y = 2^x$ only when $x$ is rational, we will get a series of dots rather than an unbroken curve. One dot above each rational $x$ on the $x$-axis. The dots will be so close together that it will be hard to tell by just looking that our graph of dots and the unbroken curve that you would usually sketch for $y = 2^x$ are slightly different. We then assume that the values of $2^x$ when $x$ is irrational are exactly those values that "fill the spaces" between the dots on our graph to make the unbroken curve of $y = 2^x$ look "the same" as the broken curve of dots. So, the value of $2^{\sqrt{3}}$ is "between" the values of $2^p$ and $2^q$ where $p$ is a rational number a teeny bit less than $\sqrt{3}$ and $q$ is a rational number a teeny bit more than $\sqrt{3}$.

This is a rather loose explanation of what we do to define irrational powers, but it is essentially correct. What we actually do is use the idea of limits and we can illustrate this idea by trying to find the value of $2^0$. We do this by looking at $2^{\frac{1}{m}}$ but we try to make this expression as close to $2^0$ as possible by making $\frac{1}{m}$ as close to 0 as possible.

And we do that by making $m$ either very big and positive or very big and negative. Let's start with $m$ being very big and positive: then $2^{\frac{1}{m}} = \sqrt[m]{2}$ and if you play around with various roots of 2, you will see that as $m$ gets bigger [and so $\frac{1}{m}$ gets smaller and $2^{\frac{1}{m}}$ gets closer to $2^0$] that the value of $\sqrt[m]{2}$ gets close to 1. Similarly, if we look at $2^{\frac{1}{m}}$ when $m$ is large and negative, we see that the value of $2^{\frac{1}{m}}$ also approaches 1 [the case when $m$ is negative takes a little more care to deal with; have a think about how it works]. This all suggest that we define $2^0 = 1$. And this idea extends to all $a^0$ where $a$ is positive. You can also sketch the graphs [a good website to use to help you understand graph sketching is DESMOS GRAPHING] of $y = a^x$ for various positive values of $a$ to see how things fit together [and notice how the shape of the graph changes depending on whether $a > 1$ or $0 < a < 1$].

Case 2

What happens when $a$ is negative?

In simple terms, things go wrong very quickly. Consider the value of $(-64)^{\frac{1}{3}}$ our definitions suggest this is just the cube root of $-64$ which is $-4$ so all seems well. Now, consider $(-64)^{\frac{1}{2}}$; this is supposed to be the square root of $-64$, but $-64$ does not have a square root, or at least it does not have one in the real number system. So, we see that it gets messy and for different values of $x$, even ones very very close together, we encounter problems. This is why, when you first meet them, index laws are only used for positive $a$ values and rational powers [although we can cope with irrational powers as we saw above].

Something to think about: what happens to $a^m$ when $a = 0$?