# 2.5 Rules of 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 the product and quotient rules for exponents, understand the meaning of zero as an exponent.

## Overview

• The Product Rule for Exponents
• The Quotient Rule for Exponents
• Zero as an Exponent

We will begin our study of the rules of exponents by recalling the definition of exponents.

## Definition of exponents

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

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

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

## Base exponent power

In ${x}^{n}$ ,

$x$ is the base
$n$ is the exponent
The number represented by ${x}^{n}$ is called a power .

The term ${x}^{n}$ is read as " $x$ to the $n$ th."

## The product rule for exponents

The first rule we wish to develop is the rule for multiplying two exponential quantities having the same base and natural number exponents. The following examples suggest this rule:

$\begin{array}{l}\begin{array}{ccccccccccc}{x}^{2}& \cdot & {x}^{4}& =& \underset{}{\underbrace{xx}}& \cdot & \underset{}{\underbrace{xxxx}}& =& \underset{}{\underbrace{xxxxxx}}& =& {x}^{6}\\ & & & & \text{\hspace{0.17em}}2& +& 4& =& 6& & \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{factors}\begin{array}{ccccc}& & & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{factors}& \end{array}\end{array}$

$\begin{array}{l}\begin{array}{ccccccccccc}a& \cdot & {a}^{2}& =& \underset{}{\underbrace{a}}& \cdot & \underset{}{\underbrace{aa}}& =& \underset{}{\underbrace{aaa}}& =& {a}^{3}\\ & & & & 1& +& 2& =& 3& & \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{cccc}\text{factors}& & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{factors}& \end{array}\end{array}$

## Product rule for exponents

If $x$ is a real number and $n$ and $m$ are natural numbers,

${x}^{n}{x}^{m}={x}^{n+m}$

To multiply two exponential quantities having the same base, add the exponents. Keep in mind that the exponential quantities being multiplied must have the same base for this rule to apply.

## Sample set a

Find the following products. All exponents are natural numbers.

${x}^{3}\cdot {x}^{5}={x}^{3+5}={x}^{8}$

${a}^{6}\cdot {a}^{14}={a}^{6+14}={a}^{20}$

${y}^{5}\cdot y={y}^{5}\cdot {y}^{1}={y}^{5+1}={y}^{6}$

${\left(x-2y\right)}^{8}{\left(x-2y\right)}^{5}={\left(x-2y\right)}^{8+5}={\left(x-2y\right)}^{13}$

$\begin{array}{ll}{x}^{3}{y}^{4}\ne {\left(xy\right)}^{3+4}\hfill & \text{Since}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{bases}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{not}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same,}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\hfill \\ \hfill & \text{product}\text{\hspace{0.17em}}\text{rule}\text{\hspace{0.17em}}\text{does}\text{\hspace{0.17em}}\text{not}\text{\hspace{0.17em}}\text{apply}\text{.}\hfill \end{array}$

## Practice set a

Find each product.

${x}^{2}\cdot {x}^{5}$

${x}^{2+5}={x}^{7}$

${x}^{9}\cdot {x}^{4}$

${x}^{9+4}={x}^{13}$

${y}^{6}\cdot {y}^{4}$

${y}^{6+4}={y}^{10}$

${c}^{12}\cdot {c}^{8}$

${c}^{12+8}={c}^{20}$

${\left(x+2\right)}^{3}\cdot {\left(x+2\right)}^{5}$

${\left(x+2\right)}^{3+5}={\left(x+2\right)}^{8}$

## Sample set b

We can use the first rule of exponents (and the others that we will develop) along with the properties of real numbers.

$2{x}^{3}\cdot 7{x}^{5}=\begin{array}{||}\hline 2\cdot 7\cdot {x}^{3+5}\\ \hline\end{array}=14{x}^{8}$

We used the commutative and associative properties of multiplication. In practice, we use these properties “mentally” (as signified by the drawing of the box). We don’t actually write the second step.

$4{y}^{3}\cdot 6{y}^{2}=\begin{array}{||}\hline 4\cdot 6\cdot {y}^{3+2}\\ \hline\end{array}=24{y}^{5}$

$9{a}^{2}{b}^{6}\left(8a{b}^{4}2{b}^{3}\right)=\begin{array}{||}\hline 9\cdot 8\cdot 2{a}^{2+1}{b}^{6+4+3}\\ \hline\end{array}=144{a}^{3}{b}^{13}$

$5{\left(a+6\right)}^{2}\cdot 3{\left(a+6\right)}^{8}=\begin{array}{||}\hline 5\cdot 3{\left(a+6\right)}^{2+8}\\ \hline\end{array}=15{\left(a+6\right)}^{10}$

$4{x}^{3}\cdot 12\cdot {y}^{2}=48{x}^{3}{y}^{2}$

The bases are the same, so we add the exponents. Although we don’t know exactly what number is, the notation indicates the addition.

## Practice set b

Perform each multiplication in one step.

$3{x}^{5}\cdot 2{x}^{2}$

$6{x}^{7}$

$6{y}^{3}\cdot 3{y}^{4}$

$18{y}^{7}$

$4{a}^{3}{b}^{2}\cdot 9{a}^{2}b$

$36{a}^{5}{b}^{3}$

${x}^{4}\cdot 4{y}^{2}\cdot 2{x}^{2}\cdot 7{y}^{6}$

$56{x}^{6}{y}^{8}$

${\left(x-y\right)}^{3}\cdot 4{\left(x-y\right)}^{2}$

$4{\left(x-y\right)}^{5}$

$8{x}^{4}{y}^{2}x{x}^{3}{y}^{5}$

$8{x}^{8}{y}^{7}$

$2aa{a}^{3}\left(a{b}^{2}{a}^{3}\right)b6a{b}^{2}$

$12{a}^{10}{b}^{5}$

${a}^{n}\cdot {a}^{m}\cdot {a}^{r}$

${a}^{n+m+r}$

## The quotient rule for exponents

The second rule we wish to develop is the rule for dividing two exponential quantities having the same base and natural number exponents.
The following examples suggest a rule for dividing two exponential quantities having the same base and natural number exponents.

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