5.2: Use Multiplication Properties of Exponents

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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

Note

Before you get started, take this readiness quiz.

Simplify: $\frac{3}{4}\cdot \frac{3}{4}$
If you missed this problem, review Exercise 1.6.13.
Simplify: $(−2)(−2)(−2)$.
If you missed this problem, review Exercise 1.5.13.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, $2^4$ means the product of $4$ factors of $2$, so $2^4$ means $2·2·2·2$.

Let’s review the vocabulary for expressions with exponents.

EXPONENTIAL NOTATION

$This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.$

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.

$This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.$

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

Example $\PageIndex{1}$

Simplify:

$4^{3}$
$7^{1}$
$\left(\frac{5}{6}\right)^{2}$
$(0.63)^{2}$

Answer

$\begin{array}{ll} & 4^{3}\\ {\text { Multiply three factors of } 4 .} & {4 \cdot 4 \cdot 4} \\ {\text { Simplify. }} & {64}\end{array}$
$\begin{array}{ll} & 7^{1}\\ \text{Multiply one factor of 7.} & 7\end{array}$
$\begin{array}{ll} &\left(\frac{5}{6}\right)^{2}\\ {\text { Multiply two factors. }} & {\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)} \\ {\text { Simplify. }} & {\frac{25}{36}}\end{array}$
$\begin{array}{ll} &(0.63)^{2}\\ {\text { Multiply two factors. }} & {(0.63)(0.63)} \\ {\text { Simplify. }} & {0.3969}\end{array}$

Example $\PageIndex{2}$

Simplify:

$6^{3}$
$15^{1}$
$\left(\frac{3}{7}\right)^{2}$
$(0.43)^{2}$

Answer

216
15
$\frac{9}{49}$
0.1849

Example $\PageIndex{3}$

Simplify:

$2^{5}$
$21^{1}$
$\left(\frac{2}{5}\right)^{3}$
$(0.218)^{2}$

Answer

32
21
$\frac{8}{125}$
0.047524

Example $\PageIndex{4}$

Simplify:

$(-5)^{4}$
$-5^{4}$

Answer

$\begin{array}{ll} &(-5)^{4}\\{\text { Multiply four factors of }-5} & {(-5)(-5)(-5)} \\ {\text { Simplify. }} & {625}\end{array}$
$\begin{array}{ll} &-5^{4}\\{\text { Multiply four factors of } 5 .} & {-(5 \cdot 5 \cdot 5 \cdot 5)} \\ {\text { Simplify. }} & {-625}\end{array}$

Notice the similarities and differences in Example $\PageIndex{4}$ part 1 and Example $\PageIndex{4}$ part 2! Why are the answers different? As we follow the order of operations in part 1 the parentheses tell us to raise the $(−5)$ to the 4^th power. In part 2 we raise just the $5$ to the 4^th power and then take the opposite.

Example $\PageIndex{5}$

Simplify:

$(-3)^{4}$
$-3^{4}$

Answer

81
−81

Example $\PageIndex{6}$

Simplify:

$(-13)^{4}$
$-13^{4}$

Answer

169
−169

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.

	$x squared times x cubed.$
What does this mean? How many factors altogether?	$x times x, multiplied by x times x. x times x has two factors. x times x times x has three factors. 2 plus 3 is five factors.$
So, we have	$x to the fifth power.$
Notice that 5 is the sum of the exponents, 2 and 3.	$x squared times x cubed is x to the power of 2 plus 3, or x to the fifth power.$

We write: \[\begin{array}{c}{x^{2} \cdot x^{3}} \\ {x^{2+3}} \\ {x^{5}}\end{array}\]

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

\[a^{m} \cdot a^{n}=a^{m+n}\]

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

\[\begin{array}{rll} {2^3\cdot2^2} &\stackrel{?}{=} & 2^{2+3}\\ {4\cdot 8} &\stackrel{?}{=} & 2^{5} \\ {32} &=& 32\checkmark\end{array}\]

Example $\PageIndex{7}$

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

Answer

	$y to the fifth power times y to the sixth power.$
Use the product property, $a^{m} \cdot a^{n}=a^{m+n}$.	$y to the power of 5 plus 6.$
Simplify.	$y to the eleventh power.$

Example $\PageIndex{8}$

Simplify: $b^{9} \cdot b^{8}$

Answer: $b^{17}$

Example $\PageIndex{9}$

Simplify: $x^{12} \cdot x^{4}$

Answer: $x^{16}$

Example $\PageIndex{10}$

Simplify:

$2^{5} \cdot 2^{9}$
$3\cdot 3^{4}$

Answer

	$2 to the fifth power times 2 to the ninth power.$
Use the product property, $a^{m} \cdot a^{n}=a^{m+n}$.	$2 to the power of 5 plus 9.$
Simplify.	$2 to the 14th power.$

	$3 to the fifth power times 3 to the fourth power.$
Use the product property, $a^{m} \cdot a^{n}=a^{m+n}$.	$3 to the power of 5 plus 4.$
Simplify.	$3 to the ninth power.$

Example $\PageIndex{11}$

Simplify:

$5\cdot 5^{5}$
$4^{9} \cdot 4^{9}$

Answer

$5^{6}$
$4^{18}$

Example $\PageIndex{12}$

Simplify:

$7^{6} \cdot 7^{8}$
$10 \cdot 10^{10}$

Answer

$7^{14}$
$10^{11}$

Example $\PageIndex{13}$

Simplify:

$a^{7} \cdot a$
$x^{27} \cdot x^{13}$

Answer

	$a to the seventh power times a.$
Rewrite, $a = a^1$	$a to the seventh power times a to the first power.$
Use the product property, $a^m\cdot a^n = a^{m+n}$.	$a to the power of 7 plus 1.$
Simplify.	$a to the eighth power.$

	$x to the twenty-seventh power times x to the thirteenth power.$
Notice, the bases are the same, so add the exponents.	$x to the power of 27 plus 13.$
Simplify.	$x to the fortieth power.$

Example $\PageIndex{14}$

Simplify:

$p^{5} \cdot p$
$y^{14} \cdot y^{29}$

Answer

$p^{6}$
$y^{43}$

Example $\PageIndex{15}$

Simplify:

$z \cdot z^{7}$
$b^{15} \cdot b^{34}$

Answer

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

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

Example $\PageIndex{16}$

Simplify: $d^{4} \cdot d^{5} \cdot d^{2}$

Answer

	$d to the fourth power times d to the fifth power times d squared.$
Add the exponents, since bases are the same.	$d to the power of 4 plus 5 plus 2.$
Simplify.	$d to the eleventh power.$

Example $\PageIndex{17}$

Simplify: $x^{6} \cdot x^{4} \cdot x^{8}$

Answer: $x^{18}$

Example $\PageIndex{18}$

Simplify: $b^{5} \cdot b^{9} \cdot b^{5}$

Answer: $b^{19}$

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.

	$x squared, in parentheses, cubed.$
What does this mean? How many factors altogether?	$x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.$
So we have	$x to the sixth power.$
Notice that 6 is the product of the exponents, 2 and 3.	$x squared cubed is x to the power of 2 times 3, or x to the sixth power.$

We write:

\[\begin{array}{c}{\left(x^{2}\right)^{3}} \\ {x^{2 \cdot 3}} \\ {x^{6}}\end{array}\]

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

\[\left(a^{m}\right)^{n}=a^{m \cdot n}\]

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

An example with numbers helps to verify this property.

\[\begin{array} {lll} \left(3^{2}\right)^{3} &\stackrel{?}{=}&3^{2 \cdot 3} \\(9)^{3} &\stackrel{?}{=} & 3^{6} \\ 729 &=&729\checkmark \end{array}\]

Example $\PageIndex{19}$

Simplify:

$\left(y^{5}\right)^{9}$
$\left(4^{4}\right)^{7}$

Answer

	$y to the fifth power, in parentheses, to the ninth power.$
Use the power property, $\big(a^m\big)^n = a^{m\cdot n}$.	$y to the power of 5 times 9.$
Simplify.	$y to the 45th power.$

	$4 to the fourth power, in parentheses, to the 7th power.$
Use the power property.	$4 to the power of 4 times 7.$
Simplify.	$4 to the twenty-eighth power.$

Example $\PageIndex{20}$

Simplify:

$ \left(b^{7}\right)^{5} $
$\left(5^{4}\right)^{3}$

Answer

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

Example $\PageIndex{21}$

Simplify:

$\left(z^{6}\right)^{9}$
$\left(3^{7}\right)^{7}$

Answer

$z^{54}$
$3^{49}$

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?

$\begin{array}{ll}{\text { What does this mean? }} & {\text { (2x) }^{3}} \\ {\text { We group the like factors together. }} & {2 x \cdot 2 x \cdot 2 x} \\ {\text { How many factors of } 2 \text { and of } x ?} & {2 \cdot 2 \cdot x^{3}} \\ {\text { Notice that each factor was raised to the power and }(2 x)^{3} \text { is } 2^{3} \cdot x^{3}}\end{array}$

$\begin{array}{ll}\text{We write:} & {(2 x)^{3}} \\ & {2^{3} \cdot x^{3}}\end{array}$

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

\[(a b)^{m}=a^{m} b^{m}\]

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

An example with numbers helps to verify this property:

\[\begin{array}{lll}(2 \cdot 3)^{2} &\stackrel{?}{=}&2^{2} \cdot 3^{2} \\ 6^{2} &\stackrel{?}{=}&4 \cdot 9 \\ 36 &=&36
\checkmark \end{array}\]

Example $\PageIndex{22}$

Simplify:

$(-9 d)^{2}$
$(3mn)^{3}$.

Answer

	$Negative 9 d squared.$
Use Power of a Product Property, $(ab)^m=a^m b^m$.	$negative 9 squared d squared.$
Simplify.	$81 d squared.$

	$3 m n cubed.$
Use Power of a Product Property, $(ab)^m=a^m b^m$.	$3 cubed m cubed n cubed.$
Simplify.	$27 m cubed n cubed.$

Example $\PageIndex{23}$

Simplify:

$(-12 y)^{2}$
$(2 w x)^{5}$

Answer

$144y^{2}$
$32w^{5} x^{5}$

Example $\PageIndex{24}$

Simplify:

$(5 w x)^{3}$
$(-3 y)^{3}$

Answer

125$w^{3} x^{3}$
$-27 y^{3}$

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

\[\begin{array}{llll} \textbf{Product Property } & a^{m} \cdot a^{n}&=&a^{m+n} \\ \textbf {Power Property } &\left(a^{m}\right)^{n}&=&a^{m n} \\ \textbf {Product to a Power } &(a b)^{m}&=&a^{m} b^{m} \end{array}\]

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

Example $\PageIndex{25}$

Simplify:

$\left(y^{3}\right)^{6}\left(y^{5}\right)^{4}$
$\left(-6 x^{4} y^{5}\right)^{2}$

Answer

$\begin{array}{ll}& \left(y^{3}\right)^{6}\left(y^{5}\right)^{4}\\ {\text { Use the Power Property. }}& y^{18} \cdot y^{20} \\ {\text { Add the exponents. }} & y^{38} \end{array}$
$\begin{array}{ll}& \left(-6 x^{4} y^{5}\right)^{2}\\ {\text { Use the Product to a Power Property. }} & {(-6)^{2}\left(x^{4}\right)^{2}\left(y^{5}\right)^{2}} \\ {\text { Use the Power Property. }} & {(-6)^{2}\left(x^{8}\right)\left(y^{10}\right)^{2}} \\ {\text { Simplify. }} & {36 x^{8} y^{10}}\end{array}$

Example $\PageIndex{26}$

Simplify:

$\left(a^{4}\right)^{5}\left(a^{7}\right)^{4}$
$\left(-2 c^{4} d^{2}\right)^{3}$

Answer

$a^{48}$
$-8 c^{12} d^{6}$

Example $\PageIndex{27}$

Simplify:

$\left(-3 x^{6} y^{7}\right)^{4}$
$\left(q^{4}\right)^{5}\left(q^{3}\right)^{3}$

Answer

81$x^{24} y^{28}$
$q^{29}$

Example $\PageIndex{28}$

Simplify:

$(5 m)^{2}\left(3 m^{3}\right)$
$\left(3 x^{2} y\right)^{4}\left(2 x y^{2}\right)^{3}$

Answer

$\begin{array}{ll}& (5 m)^{2}\left(3 m^{3}\right)\\{\text { Raise } 5 m \text { to the second power. }} & {5^{2} m^{2} \cdot 3 m^{3}} \\ {\text { Simplify. }} & {25 m^{2} \cdot 3 m^{3}} \\ {\text { Use the Commutative Property. }} & {25 \cdot 3 \cdot m^{2} \cdot m^{3}} \\ {\text { Multiply the constants and add the exponents. }} & {75 m^{5}}\end{array}$
$\begin{array}{ll} & \left(3 x^{2} y\right)^{4}\left(2 x y^{2}\right)^{3} \\ \text{Use the Product to a Power Property.} & \left(3^{4} x^{8} y^{4}\right)\left(2^{3} x^{3} y^{6}\right)\\\text{Simplify.} & \left(81 x^{8} y^{4}\right)\left(8 x^{3} y^{6}\right)\\ \text{Use the Commutative Property.} &81\cdot 8 \cdot x^{8} \cdot x^{3} \cdot y^{4} \cdot y^{6} \\\text{Multiply the constants and add the exponents.} & 648x^{11} y^{10}\\ \end{array}$

Example $\PageIndex{29}$

Simplify:

$(5 n)^{2}\left(3 n^{10}\right)$
$\left(c^{4} d^{2}\right)^{5}\left(3 c d^{5}\right)^{4}$

Answer

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

Example $\PageIndex{30}$

Simplify:

$\left(a^{3} b^{2}\right)^{6}\left(4 a b^{3}\right)^{4}$
$(2 x)^{3}\left(5 x^{7}\right)$

Answer

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

Multiply Monomials

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

Example $\PageIndex{31}$

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

Answer: $\begin{array}{ll} & \left(3 x^{2}\right)\left(-4 x^{3}\right)\\ \text{Use the Commutative Property to rearrange the terms.} & 3\cdot(-4) \cdot x^{2} \cdot x^{3}\\
\text{Multiply.} & -12 x^{5}\end{array}$

Example $\PageIndex{32}$

Multiply: $\left(5 y^{7}\right)\left(-7 y^{4}\right)$

Answer: $-35 y^{11}$

Example $\PageIndex{33}$

Multiply: $\left(-6 b^{4}\right)\left(-9 b^{5}\right)$

Answer: 54$b^{9}$

Example $\PageIndex{34}$

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

Answer: $\begin{array}{ll} & \left(\frac{5}{6} x^{3} y\right)\left(12 x y^{2}\right)\\ \text{Use the Commutative Property to rearrange the terms.} & \frac{5}{6} \cdot 12 \cdot x^{3} \cdot x \cdot y \cdot y^{2}\\ \text{Multiply.} &10x^{4} y^{3}\end{array}$

Example $\PageIndex{35}$

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

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

Example $\PageIndex{36}$

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

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

Note

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

Multiplication Properties of Exponents

Key Concepts

Exponential Notation
$This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m powder means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.$
Properties of Exponents
- If $a$ and $b$ are real numbers and $m$ and $n$ are whole numbers, then

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	$y to the fifth power times y to the sixth power.$
Use the product property, \(a^{m} \cdot a^{n}=a^{m+n}\).	$y to the power of 5 plus 6.$
Simplify.	$y to the eleventh power.$

	$a to the seventh power times a.$
Rewrite, \(a = a^1\)	$a to the seventh power times a to the first power.$
Use the product property, \(a^m\cdot a^n = a^{m+n}\).	$a to the power of 7 plus 1.$
Simplify.	$a to the eighth power.$

	$y to the fifth power, in parentheses, to the ninth power.$
Use the power property, \(\big(a^m\big)^n = a^{m\cdot n}\).	$y to the power of 5 times 9.$
Simplify.	$y to the 45th power.$

	$Negative 9 d squared.$
Use Power of a Product Property, \((ab)^m=a^m b^m\).	$negative 9 squared d squared.$
Simplify.	$81 d squared.$