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We will begin our study of the rules of exponents by recalling the definition of 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:
$$\begin{array}{l}\begin{array}{ccccccccccc}{x}^{2}& \cdot & {x}^{4}& =& \underset{}{\underbrace{xx}}& \cdot & \underset{}{\underbrace{xxxx}}& =& \underset{}{\underbrace{xxxxxx}}& =& {x}^{6}\\ & & & & \text{\hspace{0.17em}}2& +& 4& =& 6& & \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{factors}\begin{array}{ccccc}& & & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{factors}& \end{array}\end{array}$$
$$\begin{array}{l}\begin{array}{ccccccccccc}a& \cdot & {a}^{2}& =& \underset{}{\underbrace{a}}& \cdot & \underset{}{\underbrace{aa}}& =& \underset{}{\underbrace{aaa}}& =& {a}^{3}\\ & & & & 1& +& 2& =& 3& & \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{cccc}\text{factors}& & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{factors}& \end{array}\end{array}$$
Find the following products. All exponents are natural numbers.
${x}^{3}\cdot {x}^{5}={x}^{3+5}={x}^{8}$
${a}^{6}\cdot {a}^{14}={a}^{6+14}={a}^{20}$
${y}^{5}\cdot y={y}^{5}\cdot {y}^{1}={y}^{5+1}={y}^{6}$
${(x-2y)}^{8}{(x-2y)}^{5}={(x-2y)}^{8+5}={(x-2y)}^{13}$
$\begin{array}{ll}{x}^{3}{y}^{4}\ne {(xy)}^{3+4}\hfill & \text{Since}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{bases}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{not}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same,}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\hfill \\ \hfill & \text{product}\text{\hspace{0.17em}}\text{rule}\text{\hspace{0.17em}}\text{does}\text{\hspace{0.17em}}\text{not}\text{\hspace{0.17em}}\text{apply}\text{.}\hfill \end{array}$
Find each product.
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.
$4{y}^{3}\cdot 6{y}^{2}=\begin{array}{||}\hline 4\cdot 6\cdot {y}^{3+2}\\ \hline\end{array}=24{y}^{5}$
$9{a}^{2}{b}^{6}(8a{b}^{4}2{b}^{3})=\begin{array}{||}\hline 9\cdot 8\cdot 2{a}^{2+1}{b}^{6+4+3}\\ \hline\end{array}=144{a}^{3}{b}^{13}$
$5{(a+6)}^{2}\cdot 3{(a+6)}^{8}=\begin{array}{||}\hline 5\cdot 3{(a+6)}^{2+8}\\ \hline\end{array}=15{(a+6)}^{10}$
$4{x}^{3}\cdot 12\cdot {y}^{2}=48{x}^{3}{y}^{2}$
The bases are the same, so we add the exponents. Although we don’t know exactly what number
is, the notation
indicates the addition.
Perform each multiplication in one step.
${x}^{4}\cdot 4{y}^{2}\cdot 2{x}^{2}\cdot 7{y}^{6}$
$56{x}^{6}{y}^{8}$
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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