# 0.2 Exponentials

 Page 1 / 2

## 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 ${n}^{\mathrm{th}}$ 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,

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

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

To convince yourself of why this is true, use the fourth exponential law above (division of exponents) and consider what happens when $m=n$ .

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

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

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

general equation for photosynthesis
6CO2 + 6H2O + solar energy= C6H1206+ 6O2
Anastasiya
meaning of amino Acids
a diagram of an adult mosquito
what are white blood cells
white blood cell is part of the immune system. that help fight the infection.
MG
Mlungisi
Cells with a similar function, form a tissue. For example the nervous tissue is composed by cells:neurons and glia cells. Muscle tissue, is composed by different cells.
Anastasiya
I need further explanation coz celewi anything guys,,,
hey guys
Isala
on what?
Anastasiya
hie
Lish
is air homogenous or hetrogenous
homogenous
Kevin
why saying homogenous?
Isala
explain if oxygen is necessary for photosynthesis
explain if oxygen is necessary for photosynthesis
Yes, the plant does need oxygen. The plant uses oxygen, water, light, and produced food. The plant use process called photosynthesis.
MG
By using the energy of sunlight, plants convert carbon dioxide and water into carbohydrates and oxygen by photosynthesis. This happens during the day and sunlight is needed.
NOBLE
no. it s a product of the process
Anastasiya
yet still is it needed?
NOBLE
no. The reaction is: 6CO2+6H20+ solar energy =C6H12O6(glucose)+602. The plant requires Carbon dioxyde, light, and water Only, and produces glucose and oxygen( which is a waste).
Anastasiya
what was the question
joining
Godfrey
the specific one
NOBLE
the study of non and living organism is called.
Godfrey
Is call biology
Alohan
yeah
NOBLE
yes
Usher
what Is ecology
what is a cell
A cell is a basic structure and functional unit of life
Ndongya
what is biolgy
is the study of living and non living organisms
Ahmed
may u draw the female organ
i dont understand
Asal
:/
Asal
me too
DAVID
anabolism and catabolism
Anabolism refers to the process in methabolism in which complex molecules are formed "built" and requires energy to happen. Catabolism is the opposite process: complex molecules are deconstructed releasing energy, such as during glicolysis.
Anastasiya
Explain briefly independent assortment gene .
hi
Amargo
hi I'm Anatalia
Joy
what do you mean by pituitary gland
Digambar
Got questions? Join the online conversation and get instant answers!