# 0.2 Exponentials

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## Introduction

In this chapter, you will learn about the short cuts to writing $2×2×2×2$ . This is known as writing a number in exponential notation .

## Definition

Exponential notation is a short way of writing the same number multiplied by itself many times. For example, instead of $5×5×5$ , we write ${5}^{3}$ to show that the number 5 is multiplied by itself 3 times and we say “5 to the power of 3”. Likewise ${5}^{2}$ is $5×5$ and ${3}^{5}$ is $3×3×3×3×3$ . We will now have a closer look at writing numbers using exponential notation.

Exponential Notation

Exponential notation means a number written like

${a}^{n}$

where $n$ is an integer and $a$ can be any real number. $a$ is called the base and $n$ is called the exponent or index .

The $n$ th power of $a$ is defined as:

${a}^{n}=a×a×\cdots ×a\phantom{\rule{2.em}{0ex}}\left(\mathrm{n times}\right)$

with $a$ multiplied by itself $n$ times.

We can also define what it means if we have a negative exponent $-n$ . Then,

${a}^{-n}=\frac{1}{a×a×\cdots ×a\phantom{\rule{2.em}{0ex}}\left(n\mathrm{times}\right)}$
Exponentials

If $n$ is an even integer, then ${a}^{n}$ will always be positive for any non-zero real number $a$ . For example, although $-2$ is negative, ${\left(-2\right)}^{2}=-2×-2=4$ is positive and so is ${\left(-2\right)}^{-2}=\frac{1}{-2×-2}=\frac{1}{4}$ .

## Laws of exponents

There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been seen in earlier grades, but we will list all the laws here for easy reference and explain each law in detail, so that you can understand them and not only remember them.

$\begin{array}{ccc}\hfill {a}^{0}& =& 1\hfill \\ \hfill {a}^{m}×{a}^{n}& =& {a}^{m+n}\hfill \\ \hfill {a}^{-n}& =& \frac{1}{{a}^{n}}\hfill \\ \hfill {a}^{m}÷{a}^{n}& =& {a}^{m-n}\hfill \\ \hfill {\left(ab\right)}^{n}& =& {a}^{n}{b}^{n}\hfill \\ \hfill {\left({a}^{m}\right)}^{n}& =& {a}^{mn}\hfill \end{array}$

## Exponential law 1: ${a}^{0}=1$

Our definition of exponential notation shows that

$\begin{array}{ccc}\hfill {a}^{0}& =& 1\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}\left(a\ne 0\right)\hfill \end{array}$

For example, ${x}^{0}=1$ and ${\left(1\phantom{\rule{0.277778em}{0ex}}000\phantom{\rule{0.277778em}{0ex}}000\right)}^{0}=1$ .

## Application using exponential law 1: ${a}^{0}=1,\left(a\ne 0\right)$

1. ${16}^{0}$
2. $16{a}^{0}$
3. ${\left(16+a\right)}^{0}$
4. ${\left(-16\right)}^{0}$
5. $-{16}^{0}$

## Exponential law 2: ${a}^{m}×{a}^{n}={a}^{m+n}$

Our definition of exponential notation shows that

$\begin{array}{ccc}\hfill {a}^{m}×{a}^{n}& =& 1×a×...×a\phantom{\rule{2.em}{0ex}}\left(m\mathrm{times}\right)\hfill \\ \hfill & & \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}×1×a×...×a\phantom{\rule{2.em}{0ex}}\left(n\mathrm{times}\right)\hfill \\ \hfill & =& 1×a×...×a\phantom{\rule{2.em}{0ex}}\left(m+n\mathrm{times}\right)\hfill \\ \hfill & =& {a}^{m+n}\hfill \end{array}$

For example,

$\begin{array}{ccc}\hfill {2}^{7}×{2}^{3}& =& \left(2×2×2×2×2×2×2\right)×\left(2×2×2\right)\hfill \\ & =& {2}^{7+3}\hfill \\ & =& {2}^{10}\hfill \end{array}$

## Interesting fact

This simple law is the reason why exponentials were originally invented. In the days before calculators, all multiplication had to be done by hand with a pencil and a pad of paper. Multiplication takes a very long time to do and is very tedious. Adding numbers however, is very easy and quick to do. If you look at what this law is saying you will realise that it means that adding the exponents of two exponential numbers (of the same base) is the same as multiplying the two numbers together. This meant that for certain numbers, there was no need to actually multiply the numbers together in order to find out what their multiple was. This saved mathematicians a lot of time, which they could use to do something more productive.

## Application using exponential law 2: ${a}^{m}×{a}^{n}={a}^{m+n}$

1. ${x}^{2}·{x}^{5}$
2. ${2}^{3}.{2}^{4}$ [Take note that the base (2) stays the same.]
3. $3×{3}^{2a}×{3}^{2}$

## Exponential law 3: ${a}^{-n}=\frac{1}{{a}^{n}},\phantom{\rule{1.em}{0ex}}a\ne 0$

Our definition of exponential notation for a negative exponent shows that

$\begin{array}{ccc}\hfill {a}^{-n}& =& 1÷a÷...÷a\phantom{\rule{2.em}{0ex}}\left(n\mathrm{times}\right)\hfill \\ \hfill & =& \frac{1}{1×a×\cdots ×a}\phantom{\rule{2.em}{0ex}}\left(n\mathrm{times}\right)\hfill \\ \hfill & =& \frac{1}{{a}^{n}}\hfill \end{array}$

This means that a minus sign in the exponent is just another way of showing that the whole exponential number is to be divided instead of multiplied.

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