10.2 Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, $2 to the power 4$ means to multiply four factors of $2,$ so $2 to the power 4$ means $2 \cdot 2 \cdot 2 \cdot 2 .$ This format is known as exponential notation.

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

On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.

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

Example 3

Simplify:

ⓐ $open parenthesis −3 close parenthesis to the power 4$
ⓑ $−3 to the power 4$

ⓐ
	$open parenthesis −3 close parenthesis to the power 4$
Multiply four factors of −3.	$open parenthesis −3 close parenthesis open parenthesis −3 close parenthesis open parenthesis −3 close parenthesis open parenthesis −3 close parenthesis$
Simplify.	$81$

ⓑ
	$−3 to the power 4$
Multiply two factors.	$- (3 \cdot 3 \cdot 3 \cdot 3)$
Simplify.	$−81$

Notice the similarities and differences in parts ⓐ and ⓑ. Why are the answers different? In part ⓐ the parentheses tell us to raise the (−3) to the 4^th power. In part ⓑ we raise only the 3 to the 4^th power and then find the opposite.

Simplify Expressions Using the Product Property of 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. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

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


What does this mean? How many factors altogether?
So, we have
Notice that 5 is the sum of the exponents, 2 and 3.
We write:	$x^{2} \cdot x^{3}$ $x^{2 + 3}$ $x to the power 5$

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

An example with numbers helps to verify this property.

$\begin{matrix} 2^{2} \cdot 2^{3} & \overset{?}{=} & 2^{2 + 3} \\ 4 \cdot 8 & \overset{?}{=} & 2^{5} \\ 32 & = & 32 ✓ \end{matrix}$

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

Simplify Expressions Using the Power Property of 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.



What does this mean? How many factors altogether?
So, we have
Notice that 6 is the product of the exponents, 2 and 3.
We write:	${(x^{2})}^{3}$ $x^{2 \cdot 3}$ $x to the power 6$

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

An example with numbers helps to verify this property.

$\begin{matrix} {(5^{2})}^{3} & \overset{?}{=} & 5^{2 \cdot 3} \\ {(25)}^{3} & \overset{?}{=} & 5^{6} \\ 15,625 & = & 15,625 ✓ \end{matrix}$

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. Look for a pattern.

	$open parenthesis 2 x close parenthesis cubed$
What does this mean?	$2 x \cdot 2 x \cdot 2 x$
We group the like factors together.	$2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x$
How many factors of 2 and of $x ?$	$2 cubed times x cubed$
Notice that each factor was raised to the power.	$open parenthesis 2 x close parenthesis cubed is 2 cubed times x cubed$
We write:	$open parenthesis 2 x close parenthesis cubed$ $2 cubed times x cubed$

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

An example with numbers helps to verify this property:

$\begin{matrix} {(2 \cdot 3)}^{2} & \overset{?}{=} & 2^{2} \cdot 3^{2} \\ 6^{2} & \overset{?}{=} & 4 \cdot 9 \\ 36 & = & 36 ✓ \end{matrix}$

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.

Example 14

Simplify: $open parenthesis 6 n close parenthesis squared open parenthesis 4 n cubed close parenthesis.$

	$open parenthesis 6 n close parenthesis squared open parenthesis 4 n cubed close parenthesis$
Raise $6 n$ to the second power.	$6 squared n squared times 4 n cubed$
Simplify.	$36 n squared times 4 n cubed$
Use the Commutative Property.	$36 \cdot 4 \cdot n^{2} \cdot n^{3}$
Multiply the constants and add the exponents.	$144 n to the power 5$

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.

Example 15

Simplify: $open parenthesis 3 p squared q close parenthesis to the power 4 open parenthesis 2 p q squared close parenthesis cubed.$

	$open parenthesis 3 p squared q close parenthesis to the power 4 open parenthesis 2 p q squared close parenthesis cubed$
Use the Power of a Product Property.	$3^{4} {(p^{2})}^{4} q^{4} \cdot 2^{3} p^{3} {(q^{2})}^{3}$
Use the Power Property.	$81 p^{8} q^{4} \cdot 8 p^{3} q^{6}$
Use the Commutative Property.	$81 \cdot 8 \cdot p^{8} \cdot p^{3} \cdot q^{4} \cdot q^{6}$
Multiply the constants and add the exponents for each variable.	$648 p^{11} q^{10}$

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Key Concepts

Exponential Notation

This is read $a$ to the $m^{th}$ power.
Product Property of Exponents
- If $a$ is a real number and $m, n$ are counting numbers, then
  $a^{m} \cdot a^{n} = a^{m + n}$
- To multiply with like bases, add the exponents.
Power Property for Exponents
- If $a$ is a real number and $m, n$ are counting numbers, then
  ${(a^{m})}^{n} = a^{m \cdot n}$
Product to a Power Property for Exponents
- If $a$ and $b$ are real numbers and $m$ is a whole number, then
  ${(a b)}^{m} = a^{m} b^{m}$

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

$4 to the power 5$

1,024

$10 cubed$

${(\frac{1}{2})}^{2}$

$\frac{1}{4}$

${(\frac{3}{5})}^{2}$

$open parenthesis 0.2 close parenthesis cubed$

0.008

$open parenthesis 0.4 close parenthesis cubed$

$open parenthesis −5 close parenthesis to the power 4$

625

$open parenthesis −3 close parenthesis to the power 5$

$−5 to the power 4$

−625

$−3 to the power 5$

$−10 to the power 4$

−10,000

$−2 to the power 6$

${(- \frac{2}{3})}^{3}$

$- \frac{8}{27}$

${(- \frac{1}{4})}^{4}$

$minus 0.5 squared$

−0.25

$- {0.1}^{4}$

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression using the Product Property of Exponents.

$x^{3} \cdot x^{6}$

x⁹

$m^{4} \cdot m^{2}$

$a \cdot a^{4}$

a⁵

$y^{12} \cdot y$

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

3¹⁴

$5^{10} \cdot 5^{6}$

$z \cdot z^{2} \cdot z^{3}$

z⁶

$a \cdot a^{3} \cdot a^{5}$

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

x^a+2

$y^{p} \cdot y^{3}$

$y^{a} \cdot y^{b}$

y^a+b

$x^{p} \cdot x^{q}$

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression using the Power Property of Exponents.

$open parenthesis u to the power 4 close parenthesis squared$

u⁸

$open parenthesis x squared close parenthesis to the power 7$

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

y²⁰

${(a^{3})}^{2}$

$open parenthesis 10 squared close parenthesis to the power 6$

10¹²

$open parenthesis 2 to the power 8 close parenthesis cubed$

${(x^{15})}^{6}$

x⁹⁰

${(y^{12})}^{8}$

${(x^{2})}^{y}$

x^2y

${(y^{3})}^{x}$

${(5^{x})}^{y}$

5^xy

${(7^{a})}^{b}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

$open parenthesis 5 a close parenthesis squared$

25a²

$open parenthesis 7 x close parenthesis squared$

$open parenthesis minus 6 m close parenthesis cubed$

−216m³

$open parenthesis minus 9 n close parenthesis cubed$

$open parenthesis 4 r s close parenthesis squared$

16r²s²

$open parenthesis 5 a b close parenthesis cubed$

$open parenthesis 4 x y z close parenthesis to the power 4$

256x⁴y⁴z⁴

$open parenthesis minus 5 a b c close parenthesis cubed$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${(x^{2})}^{4} \cdot {(x^{3})}^{2}$

x¹⁴

${(y^{4})}^{3} \cdot {(y^{5})}^{2}$

${(a^{2})}^{6} \cdot {(a^{3})}^{8}$

a³⁶

${(b^{7})}^{5} \cdot {(b^{2})}^{6}$

$open parenthesis 3 x close parenthesis squared open parenthesis 5 x close parenthesis$

45x³

$open parenthesis 2 y close parenthesis cubed open parenthesis 6 y close parenthesis$

$open parenthesis 5 a close parenthesis squared open parenthesis 2 a close parenthesis cubed$

200a⁵

$open parenthesis 4 b close parenthesis squared open parenthesis 3 b close parenthesis cubed$

$open parenthesis 2 m to the power 6 close parenthesis cubed$

8m¹⁸

$open parenthesis 3 y squared close parenthesis to the power 4$

$open parenthesis 10 x squared y close parenthesis cubed$

1,000x⁶y³

${(2 m n^{4})}^{5}$

$open parenthesis −2 a cubed b squared close parenthesis to the power 4$

16a¹²b⁸

$open parenthesis −10 u squared v to the power 4 close parenthesis cubed$

${(\frac{2}{3} x^{2} y)}^{3}$

$\frac{8}{27} x^{6} y^{3}$

${(\frac{7}{9} p q^{4})}^{2}$

$open parenthesis 8 a cubed close parenthesis squared open parenthesis 2 a close parenthesis to the power 4$

1,024a¹⁰

$open parenthesis 5 r squared close parenthesis cubed open parenthesis 3 r close parenthesis squared$

${(10 p^{4})}^{3} {(5 p^{6})}^{2}$

25,000p²⁴

${(4 x^{3})}^{3} {(2 x^{5})}^{4}$

${(\frac{1}{2} x^{2} y^{3})}^{4} {(4 x^{5} y^{3})}^{2}$

x¹⁸y¹⁸

${(\frac{1}{3} m^{3} n^{2})}^{4} {(9 m^{8} n^{3})}^{2}$

${(3 m^{2} n)}^{2} {(2 m n^{5})}^{4}$

144m⁸n²²

${(2 p q^{4})}^{3} {(5 p^{6} q)}^{2}$

Multiply Monomials

In the following exercises, multiply the following monomials.

$open parenthesis 12 x squared close parenthesis open parenthesis −5 x to the power 4 close parenthesis$

−60x⁶

$open parenthesis −10 y cubed close parenthesis open parenthesis 7 y squared close parenthesis$

$open parenthesis −8 u to the power 6 close parenthesis open parenthesis −9 u close parenthesis$

72u⁷

$open parenthesis −6 c to the power 4 close parenthesis open parenthesis −12 c close parenthesis$

$(\frac{1}{5} r^{8}) (20 r^{3})$

4r¹¹

$(\frac{1}{4} a^{5}) (36 a^{2})$

$open parenthesis 4 a cubed b close parenthesis open parenthesis 9 a squared b to the power 6 close parenthesis$

36a⁵b⁷

$(6 m^{4} n^{3}) (7 m n^{5})$

$(\frac{4}{7} x y^{2}) (14 x y^{3})$

8x²y⁵

$(\frac{5}{8} u^{3} v) (24 u^{5} v)$

$(\frac{2}{3} x^{2} y) (\frac{3}{4} x y^{2})$

$\frac{1}{2} x^{3} y^{3}$

$(\frac{3}{5} m^{3} n^{2}) (\frac{5}{9} m^{2} n^{3})$

Everyday Math

Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is $6 squared,$ on the third round is $6 cubed,$ as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.

Round	Number of people
$1$	$6$
$2$	$6 squared$
$3$	$6 cubed$
$\dots$	$\dots$
$8$	$?$

1,679,616

Salary Raul’s boss gives him a $5%$ raise every year on his birthday. This means that each year, Raul’s salary is $1.05$ times his last year’s salary. If his original salary was $$40,000$ , his salary after $1$ year was $$40,000 open parenthesis 1.05 close parenthesis,$ after $2$ years was $$40,000 open parenthesis 1.05 close parenthesis squared,$ after $3$ years was $$40,000 open parenthesis 1.05 close parenthesis cubed,$ as shown in the table below. What will Raul’s salary be after $10$ years? Simplify the expression, to show Raul’s salary in dollars.

Year	Salary
$1$	$$40,000 open parenthesis 1.05 close parenthesis$
$2$	$$40,000 open parenthesis 1.05 close parenthesis squared$
$3$	$$40,000 open parenthesis 1.05 close parenthesis cubed$
$\dots$	$\dots$
$10$	$?$

Writing Exercises

Use the Product Property for Exponents to explain why $x \cdot x = x^{2} .$

Answers will vary.

Explain why $−5 cubed equals open parenthesis −5 close parenthesis cubed$ but ${−5}^{4} \neq {(−5)}^{4} .$

Jorge thinks ${(\frac{1}{2})}^{2}$ is $1 .$ What is wrong with his reasoning?

Answers will vary.

Explain why $x^{3} \cdot x^{5}$ is $x to the power 8,$ and not $x to the power 15.$

Self Check

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

A self-assessment checklist for students to rate their understanding of simplifying expressions with exponents and multiplying monomials, using categories like 'Confidently' and 'No-I don't get it!' — **Figure 10.3**

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

ⓐ
	$5 cubed$
Multiply 3 factors of 5.	$5 \cdot 5 \cdot 5$
Simplify.	$125$

ⓑ
	$9 to the power 1$
Multiply 1 factor of 9.	$9$

	$x^{5} \cdot x^{7}$
Use the product property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.	$x to the power 12$

	$b^{4} \cdot b$
Rewrite, $b = b^{1} .$	$b^{4} \cdot b^{1}$
Use the product property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.	$b to the power 5$

	$2^{7} \cdot 2^{9}$
Use the product property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.	$2 to the power 16$

ⓐ
	${(\frac{7}{8})}^{2}$
Multiply two factors.	$(\frac{7}{8}) (\frac{7}{8})$
Simplify.	$\frac{49}{64}$

ⓑ
	$open parenthesis 0.74 close parenthesis squared$
Multiply two factors.	$open parenthesis 0.74 close parenthesis open parenthesis 0.74 close parenthesis$
Simplify.	$0.5476$

	$y^{17} \cdot y^{23}$
Notice, the bases are the same, so add the exponents.
Simplify.	$y to the power 40$

	$x^{3} \cdot x^{4} \cdot x^{2}$
Add the exponents, since the bases are the same.
Simplify.	$x to the power 9$

ⓐ
	${(x^{5})}^{7}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.	$x to the power 35$

ⓑ
	${(3^{6})}^{8}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.	$3 to the power 48$

	$open parenthesis −11 x close parenthesis squared$
Use the Power of a Product Property, ${(a b)}^{m} = a^{m} b^{m} .$
Simplify.	$121 x squared$

	$open parenthesis 3 x y close parenthesis cubed$
Raise each factor to the third power.
Simplify.	$27 x cubed y cubed$

	${(x^{2})}^{6} {(x^{5})}^{4}$
Use the Power Property.	$x^{12} \cdot x^{20}$
Add the exponents.	$x to the power 32$

	${(- 7 x^{3} y^{4})}^{2}$
Take each factor to the second power.	$open parenthesis −7 close parenthesis squared open parenthesis x cubed close parenthesis squared open parenthesis y to the power 4 close parenthesis squared$
Use the Power Property.	$49 x^{6} y^{8}$

	$open parenthesis 4 x squared close parenthesis open parenthesis minus 5 x cubed close parenthesis$
Use the Commutative Property to rearrange the factors.	$4 \cdot (−5) \cdot x^{2} \cdot x^{3}$
Multiply.	$- 20 x^{5}$

	$(\frac{3}{4} c^{3} d) (12 c d^{2})$
Use the Commutative Property to rearrange the factors.	$\frac{3}{4} \cdot 12 \cdot c^{3} \cdot c \cdot d \cdot d^{2}$
Multiply.	$9 c to the power 4 d cubed$