# 5.2 Use Multiplication Properties of Exponents

## Learning Objectives

By the end of this section, you will be able to:

• Simplify expressions with exponents
• Simplify expressions using the Product Property for Exponents
• Simplify expressions using the Power Property for Exponents
• Simplify expressions using the Product to a Power Property
• Simplify expressions by applying several properties
• Multiply monomials

## Try it

Before you get started, take this readiness quiz:

1) Simplify: $\frac{3}{4}\times\frac{3}{4}$
2) Simplify: $(-2)(-2)(-2)$.
3) What is the place value of the $6$ in the number $64,891$?
4) Name the decimal: $0.0012$.
5) Subtract: $5-(-3)$.

## Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, ${2}^{4}$means to multiply $2$ by itself $4$ times, so ${2}^{4}$ means $2\cdot 2\cdot 2\cdot 2$.

Let’s review the vocabulary for expressions with exponents.

### Exponential Notation

\begin{align*}&\;a^{m{\color{blue}{\leftarrow\text{exponent}}}}\;&\;&a^m\text{means multiply}\;m\;\text{factors of}\;a\;\;\\&{\color{red}{\underset{base}\uparrow}}\;&\;&a^m=\underbrace{a\cdot a\cdot a\cdot.....\cdot a}_{\color{blue}{{m\;\text{factors}}}}\;\;\end{align*}

This is read $a$ to the ${m}^{th}$ power.

In the expression ${a}^{m}$, the exponent $m$ tells us how many times we use the base $a$ as a factor.

\begin{align*}&\;\;\;\;\;4^3\;&\left(-9\right)^5\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\&\underbrace{4\cdot4\cdot4}_{{\color{blue}{3\;\text{factors}}}}\;&\underbrace{\left(-9\right)\left(-9\right)\left(-9\right)\left(-9\right)\left(-9\right)}_{{\color{blue}{5\;\text{factors}}}}\end{align*}

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

### Example 5.2.1

Simplify:

a. ${4}^{3}$
b. ${7}^{1}$
c. $(\frac{5}{6})^{2}$
d. $(0.63)^{2}$

Solution

a.

Step 1: Multiply three factors of $4$.

\begin{align*} &4\times{4}\times{4}\\ \text{Simplify.}\;\;&64 \end{align*}

b.

Step 1: Multiply one factor of $7$.

$7$

c.

Step 1: Multiply two factors.

\begin{align*} &\left(\frac{5}{6}\right)\times\left({\frac{5}{6}}\right)\\ \text{Simplify.}\;\;\;&\;\frac{25}{36} \end{align*}

d.

Step 1: Multiply two factors.

\begin{align*} &(0.63)\times{(0.63)}\\ \text{Simplify.}\;\;\;&0.3969 \end{align*}

### Try It

6) Simplify:

a. ${6}^{3}$
b. ${15}^{1}$
c. $(\frac{3}{7})^{2}$
d. $(0.43)^{2}$

Solution

a. $216$
b. $15$
c. $\frac{9}{49}$
d. $0.1849$

7) Simplify:

a. ${2}^{5}$
b. ${21}^{1}$
c. $(\frac{2}{5})^{3}$
d. $(0.218)^{2}$

Solution

a. $32$
b. $21$
c. $\frac{8}{125}$
d. $0.047524$

### Example 5.2.2

Simplify:

a. $(-5)^{4}$
b. ${−}{5}^{4}$

Solution

a.

Step 1: Multiply four factors of $-5$.

\begin{align*} &(-5)(-5)(-5)(-5)\\ \text{Simplify.}\;\;\;&625 \end{align*}

b.

Step 1: Multiply four factors of $5$.

\begin{align*} &-(5\times5\times5\times5)\\ \text{Simplify.}\;\;\;&6-25 \end{align*}

### Try It

8) Simplify:

a. $(-3)^{4}$
b. $−{3}^{4}$

Solution

a. $81$
b. $-81$

9) Simplify:

a. $(-13)^{2}$
b. ${−}{13}^{2}$

Solution

a. $169$
b. $-169$

Notice the similarities and differences in the “Example 5.2.2″ a. and b.! Why are the answers different? As we follow the order of operations in part a. the parentheses tell us to raise the $-5$ to the $4^{th}$ power. In part b. we raise just the $5$ to the $4^{th}$ power and then take the opposite.

## Simplify Expressions Using the Product Property for Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

We’ll derive the properties of exponents by looking for patterns in several examples.

First, we will look at an example that leads to the Product Property.

 $x^2\cdot x^3$ What does this mean? How many factors altogether? $\underbrace{\underbrace{x\cdot x}_{{\color{blue}{2\text{ factors}}}}\;\;\cdot\;\;\underbrace{x\cdot x\cdot x}_{{\color{blue}{3\text{ factors}}}}}_{{\color{blue}{5\text{ fators}}}}$ So, we have $x^5$ Notice that $5$ is the sum of the exponents, $2$ and $3$. $x^2\cdot x^3$ is $x^{2+3}$, or $x^5$

We write:

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

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

### Product Property for Exponents

If $a$ is a real number, and $m$ and $n$ are counting numbers, then

${a}^{m}\times{a}^{n}={a}^{m+n}$

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property:

$\begin{eqnarray*}2^2\;\times\;2^{3\;}& \overset?= &\;2^{2+3}\\4\;\times\;8\;& \overset?=& \;2^5\\32\;& = &\;32\\\end{eqnarray*}$

### Example 5.2.3

Simplify: ${y}^{5}\times{y}^{6}$

Solution

Step 1: Use the product property, $a^m\cdot a^n=a^{m+n}$

\begin{align*}&y^{{\color{red}{5\;+\;6}}}\\\text{Simplify.}\;\;&y^{11}\end{align*}

### Try It

10) Simplify: ${b}^{9}\times{b}^{8}$

Solution

${b}^{17}$

11) Simplify: ${x}^{12}\times{x}^{4}$.

Solution

${x}^{16}$

### Example 5.2.4

Simplify:

a. ${2}^{5}\times{2}^{9}$
b. $3\times{3}^{4}$

Solution

a.

Step 1: Use the product property, $a^m\cdot a^n=a^{m+n}$

\begin{align*}&2^{{\color{red}{5\;+\;9}}}\\\text{Simplify.}\;\;&2^{14}\end{align*}

b.

Step 1: Use the product property, $a^m\cdot a^n=a^{m+n}$

\begin{align*}&3^{{\color{red}{1\;+\;4}}}\\\text{Simplify.}\;\;&3^5\end{align*}

### Try It

12) Simplify:

a. $5\times{5}^{5}$
b. ${4}^{9}\times{4}^{9}$

Solution

a. ${5}^{6}$
b.${4}^{18}$

13) Simplify:

a. ${7}^{6}\times{7}^{8}$
b. $10\times{10}^{10}$.

Solution

a. ${7}^{14}$
b. ${10}^{11}$

### Example 5.2.5

Simplify:

a. ${a}^{7}\times{a}$
b. ${x}^{27}\times{x}^{13}$

Solution

a.
Step 1: Rewrite, $a=a^1$.

$a^7\;\times\;a^1$

Step 2: Use the product property, $a^m\cdot a^n=a^{m+n}$

\begin{align*}&3a^{{\color{red}{7\;+\;1}}}\\\text{Simplify.}\;\;&a^8\end{align*}

b.

Step 1: Notice, the bases are the same, so add the exponents.

\begin{align*}&x^{{\color{red}{27\;+\;13}}}\\\text{Simplify.}\;\;&x^{40}\end{align*}

### Try It

14) Simplify:

a. ${p}^{5}\times {p}$
b. ${y}^{14}\times{y}^{29}$

Solution

a. ${p}^{6}$
b. ${y}^{43}$

15) Simplify:

a. $z\times{z}^{7}$
b. ${b}^{15}\times{b}^{34}$.

Solution

a. ${z}^{8}$
b. ${b}^{49}$

We can extend the Product Property for Exponents to more than two factors.

### Example 5.2.6

Simplify: ${d}^{4}·{d}^{5}·{d}^{2}$

Solution

Step 1: Add the exponents, since bases are the same.

\begin{align*}&d^{\color{red}{4\;+\;5\;+\;2\;}}\\\text{Simplify.}\;\;&d^{11}\end{align*}

### Try It

16) Simplify: ${x}^{6}\times{x}^{4}\times{x}^{8}$

Solution

${x}^{18}$

17) Simplify: ${b}^{5}\times{b}^{9}\times{b}^{5}$

Solution

${b}^{19}$

## Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

 $(x^2)^3$ What does this mean? How many factors altogether? $\;\;\;\;x^2\;\;\;\;\cdot\;\;\;\;\;\;x^2\;\;\;\;\cdot\;\;\;\;\;\;x^2\\\underbrace{\underbrace{x\cdot x}_{{\color{blue}{2\text{ factors}}}}\;\;\cdot\;\;\underbrace{x\cdot x}_{{\color{blue}{2\text{ factors}}}}\;\;\cdot\;\;\underbrace{x\cdot x}_{{\color{blue}{2\text{ factors}}}}}_{{\color{blue}{6\text{ factors}}}}$ So, we have $x^6$ Notice that 6 is the product of the exponents, 2 and 3. $(x^2)^3$ is $x^{2\cdot3}$ or $x^6$

We write:

$(x^2)^3$
$x^{({2}\times{3})}$
$x^6$

We multiplied the exponents. This leads to the Power Property for Exponents.

### Power Property for Exponents

If $a$ is a real number, and $m$ and $n$ are whole numbers, then

$(a^m)^n=a^{mn}$

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property:

$\begin{eqnarray*}{(3^2)}^3\;& \overset?= &\;3^{(2\;\times\;3)}\\{(9)}^3\;& \overset?= &\;3^6\\729\;& = &\;729\\\\\end{eqnarray*}$

### Example 5.2.7

Simplify:

a.$({y}^{5})^{9}$
b. (${4}^{4})^{7}$

Solution

a.

Step 1: Use the power property, $(a^m)^n=a^{a\cdot n}$

\begin{align*}&y^{\color{red}{5\;\times\;9}}\\\text{Simplify.}\;\;&y^{45}\end{align*}

b.

Step 1: Use the power property.

\begin{align*}&4^{\color{red}{4\;\times\;7}}\\\text{Simplify.}\;\;&4^{28}\end{align*}

### Try It

18) Simplify:

a. $({b}^{7})^{5}$
b. $({5}^{4})^{3}$

Solution

a. ${b}^{35}$
b. ${5}^{12}$

19) Simplify:

a. $({z}^{6})^{9}$
b. $({3}^{7})^{7}$

Solution

a. $z^{54}$
b. $3^{49}$

## Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

 $(2x)^{3}$ What does this mean? $2x\cdot 2x\cdot 2x$ We group the like factors together. $2\cdot 2\cdot 2\cdot x\cdot x\cdot x$ How many factors of $2$ and of $x$? ${2}^{3}\times {x}^{3}$

Notice that each factor was raised to the power and $(2x)^{3}$ is ${2}^{3} \times {x}^{3}$.

We write:

$(2x)^{3}$
${2}^{3}\times {x}^{3}$

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

### Product to a Power Property for Exponents

If $a$ and $b$ are real numbers and $m$ is a whole number, then

$(ab)^{m} = {a}^{m} {b}^{m}$

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

$\begin{eqnarray*}2{(2\;\times\;3)}^2\;& \overset?= &\;2^23^2\\6^2\;& \overset?= &\;(4)(9)\\36\;& = &\;36\\\end{eqnarray*}$

### Example 5.2.8

Simplify:

a. $(-9d)^{2}$
b.  $(3mn)^{3}$

Solution

a.

Step 1: Use Power of a Product Property, $(ab)^{m} = {a}^{m} {b}^{m}$

\begin{align*}&\left(-9\right)^{\color{red}{2}}d^{\color{red}{9}}\\\text{Simplify.}\;\;&81d^2 \end{align*}

b.

Step 1: Use Power of a Product Property, $(ab)^{m} = {a}^{m} {b}^{m}$

\begin{align*}&\left(3\right)^{\color{red}{3}}m^{\color{red}{3}}n^{\color{red}{3}}\\\text{Simplify.}\;\;&27m^3n^3\end{align*}

### Try It

20) Simplify:

a. $(-12y)^{2}$
b. $(2wx)^{5}$

Solution

a. $144y^{2}$
b. $32{w}^{5}{x}^{5}$

21) Simplify:

a. ${5wx}^{3}$
b. $(-3y)^{3}$

Solution

a.$125{w}^{3}{x}^{3}$
b. $-27{y}^{3}$

## Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

### Properties of Exponents

If $a$ and $b$ are real numbers, and $m$ and $n$ are whole numbers, then

 Product Property ${a}^{m}\times{a}^{n}={a}^{m+n}$ Power Property $({a}^{m})^{n}={a}^{mn}$ Product to a Power $(ab)^{m}={a}^{m}{b}^{m}$

All exponent properties hold true for any real numbers $m$ and $n$. Right now, we only use whole-number exponents.

### Example 5.2.9

Simplify:

a. $({y}^{3})^{6}({y}^{5})^{4}$
b. $(-6{x}^{4}{y}^{5})^{2}$

Solution

a.

Step 1: Use the Power Property.

${y}^{18}\times{y}^{20}$

${y}^{38}$

b.

Step 1: Use the Product to a Power Property.

$(-6)^2({x}^4)^2(y^5)^2$

Step 2: Use the Power Property.

\begin{align*}&(-6)^{2}(x^4)^2(y^5)^2\\\text{Simplify.}\;\;&36{x}^{8}{y}^{10}\end{align*}

### Try It

22) Simplify:

a. $({a}^{4})^{5}({a}^{7})^{4}$
b. $(-2{c}^{4}{d}^{2})^{3}$

Solution

a. ${a}^{48}$
b. $(-8{c}^{12}{d}^{6})$

23) Simplify:

a. $(-3{x}^{6}{y}^{7})^{4}$
b. $({q}^{4})^{5}({q}^{3})^{3}$

Solution

a.$81{x}^{24}{y}^{28}$
b. ${q}^{29}$

### Example 5.2.10

Simplify:

a.$(5m)^{2}(3{m})^{3}$
b. $(3{x}^{2}y)^{4}(2xy^2)^{3}$

Solution

a.

Step 1: Raise $5m$ to the second power.

\begin{align*} &{5}^{2}{m}^{2}\times3{m}^{3}\\ \text{Simplify.}\;\;&25{m}^{2}\times3{m}^{3} \end{align*}

Step 2: Use the Commutative Property.

$25\times{3}\times{m}^{2}\times{m}^{3}$

Step 3: Multiply the constants and add the exponents.

$75{m}^{5}$

b.

Step 1: Use the Product to a Power Property.

\begin{align*} &({3}^{4}{x}^{8}{y}^{4})({2}^{3}{x}^{3}{y}^{6})\\ \text{Simplify.}\;\;&(81{x}^{8}{y}^{4})(8{x}^{3}{y}^{6}) \end{align*}

Step 2: Use the Commutative Property.

$81\times8\times{x}^{8}\times{x}^{3}\times{y}^{4}\times{y}^{6}$

Step 3: Multiply the constants and add the exponents.

$648{x}^{11}{y}^{10}$

### Try It

24) Simplify:

a. $(5n)^{2}(3{n}^{10})$
b. $({c}^{4}{d}^{2})^{5}(3c{d}^{5})^{4}$

Solution

a. $75{n}^{12}$
b. $81{c}^{24}{d}^{30}$

25) Simplify:

a. $({a}^{3}{b}^{2})^{6}{(4a{b}^{3})}^{4}$
b. $(2x)^{3}(5{x}^{7})$

Solution

a. $256{a}^{22}{b}^{24}$
b. $40{x}^{10}$

## Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

### Example 5.2.11

Multiply: $(3{x}^{2})(-4{x}^{3})$

Solution

Step 1: Use the Commutative Property to rearrange the terms.

\begin{align*} &3(-4){x}^{2}{x}^{3}\\ \text{Multiply.}\;\;&-12{x}^{5} \end{align*}

### Try It

26) Multiply: $(5{y}^{7})(-7{y}^{4})$

Solution

$-35{y}^{11}$

27) Multiply: $(-6{b}^{4})(-9{b}^{5})$

Solution

$54{b}^{9}$

### Example 5.2.12

Multiply: $(\frac{5}{6}{x}^{3}y)(12x{y}^{2})$

Solution

Step 1: Use the Commutative Property to rearrange the terms.

\begin{align*} &\frac{5}{6}\times{12}{x}^{3}xy{y}^{2}\\ \text{Multiply.}\;\;&10{x}^{4}{y}^{3} \end{align*}

### Try It

28) Multiply: $(\frac{2}{5}{a}^{4}{b}^{3})(15a{b}^{3})$

Solution

$6{a}^{5}{b}^{6}$

29) Multiply: $(\frac{2}{3}{r}^{5}s)(12{r}^{6}{s}^{7})$

Solution

$8{r}^{11}{s}^{8}$

Access these online resources for additional instruction and practice with using multiplication properties of exponents:

### Key Concepts

• Exponential Notation

\begin{align*}&\;a^{m{\color{blue}{\leftarrow\text{exponent}}}}\;&\;&a^m\text{means multiply}\;m\text{ factors of }a\;\;\\ &{\color{red}{\underset{base}\uparrow}}\;&\;&a^m=\underbrace{a\cdot a\cdot a\cdot.....\cdot a}_{\color{blue}{{m\text{ factors}}}}\;\;\end{align*}

• Properties of Exponents
• If $a$, and $b$ are real numbers and $m$, $n$ are whole numbers, then
• Product Property        $a^{m}a^{n} = {a}^{m+n}$
• Power Property          $({a}^m)^n={a}^{mn}$
• Product to a Power    $(ab)^m = a^mb^m$

### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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After reviewing this checklist, what will you do to become confident for all goals?