# 2.4 Exponents

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: understand exponential notation, be able to read exponential notation, understand how to use exponential notation with the order of operations.

## Overview

• Exponential Notation
• The Order of Operations

## Exponential notation

In Section [link] we were reminded that multiplication is a description for repeated addition. A natural question is “Is there a description for repeated multiplication?” The answer is yes. The notation that describes repeated multiplication is exponential notation .

## Factors

In multiplication, the numbers being multiplied together are called factors . In repeated multiplication, all the factors are the same. In nonrepeated multiplication, none of the factors are the same. For example,

$\begin{array}{ll}18\cdot 18\cdot 18\cdot 18\hfill & \text{Repeated}\text{\hspace{0.17em}}\text{multiplication}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}18.\text{\hspace{0.17em}}\text{All}\text{\hspace{0.17em}}\text{four}\text{\hspace{0.17em}}\text{factors},\text{\hspace{0.17em}}18,\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \\ x\cdot x\cdot x\cdot x\cdot x\hfill & \text{Repeated}\text{\hspace{0.17em}}\text{multiplication}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}\text{All}\text{\hspace{0.17em}}\text{five}\text{\hspace{0.17em}}\text{factors},\text{\hspace{0.17em}}x,\text{are}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \\ 3\cdot 7\cdot a\hfill & \text{Nonrepeated}\text{\hspace{0.17em}}\text{multiplication}\text{.}\text{\hspace{0.17em}}\text{None}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{factors}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \end{array}$

Exponential notation is used to show repeated multiplication of the same factor. The notation consists of using a superscript on the factor that is repeated . The superscript is called an exponent .

## Exponential notation

If $x$ is any real number and $n$ is a natural number, then

${x}^{n}=\underset{n\text{\hspace{0.17em}}\text{factors}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}x}{\underbrace{x\cdot x\cdot x\cdot ...\cdot x}}$

An exponent records the number of identical factors in a multiplication.

Note that the definition for exponential notation only has meaning for natural number exponents. We will extend this notation to include other numbers as exponents later.

## Sample set a

$7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7={7}^{6}$ .

The repeated factor is 7. The exponent 6 records the fact that 7 appears 6 times in the multiplication.

$x\cdot x\cdot x\cdot x={x}^{4}$ .

The repeated factor is $x$ . The exponent 4 records the fact that $x$ appears 4 times in the multiplication.

$\left(2y\right)\left(2y\right)\left(2y\right)={\left(2y\right)}^{3}$ .

The repeated factor is $2y$ . The exponent 3 records the fact that the factor $2y$ appears 3 times in the multiplication.

$2yyy=2{y}^{3}$ .

The repeated factor is $y$ . The exponent 3 records the fact that the factor $y$ appears 3 times in the multiplication.

$\left(a+b\right)\left(a+b\right)\left(a-b\right)\left(a-b\right)\left(a-b\right)={\left(a+b\right)}^{2}{\left(a-b\right)}^{3}$ .

The repeated factors are $\left(a+b\right)$ and $\left(a-b\right)$ , $\left(a+b\right)$ appearing 2 times and $\left(a-b\right)$ appearing 3 times.

## Practice set a

Write each of the following using exponents.

$a\cdot a\cdot a\cdot a$

${a}^{4}$

$\left(3b\right)\left(3b\right)\left(5c\right)\left(5c\right)\left(5c\right)\left(5c\right)$

${\left(3b\right)}^{2}{\left(5c\right)}^{4}$

$2\cdot 2\cdot 7\cdot 7\cdot 7\cdot \left(a-4\right)\left(a-4\right)$

${2}^{2}\cdot {7}^{3}{\left(a-4\right)}^{2}$

$8xxxyzzzzz$

$8{x}^{3}y{z}^{5}$

## Caution

It is extremely important to realize and remember that an exponent applies only to the factor to which it is directly connected.

## Sample set b

$8{x}^{3}$ means $8\cdot xxx$ and not $8x8x8x$ . The exponent 3 applies only to the factor $x$ since it is only to the factor $x$ that the 3 is connected.

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