6.2 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 2 by itself 4 times, so $2 to the power 4$ means $2 \cdot 2 \cdot 2 \cdot 2$ .

Let’s review the vocabulary for expressions with exponents.

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

Example 1

Simplify: ⓐ $4 cubed$ ⓑ $7 to the power 1$ ⓒ ${(\frac{5}{6})}^{2}$ ⓓ $open parenthesis 0.63 close parenthesis squared.$

ⓐ	$4 cubed$
Multiply three factors of 4.	$4 \cdot 4 \cdot 4$
Simplify.	$64$
ⓑ	$7 to the power 1$
Multiply one factor of 7.	$7$
ⓒ	${(\frac{5}{6})}^{2}$
Multiply two factors.	$(\frac{5}{6}) (\frac{5}{6})$
Simplify.	$\frac{25}{36}$
ⓓ	$open parenthesis 0.63 close parenthesis squared$
Multiply two factors.	$open parenthesis 0.63 close parenthesis open parenthesis 0.63 close parenthesis$
Simplify.	$0.3969$

Notice the similarities and differences in Example 2ⓐ and Example 2ⓑ ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the $open parenthesis −5 close parenthesis$ to the 4^th power. In part ⓑ 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.


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:

$\begin{matrix} x^{2} \cdot x^{3} \\ x^{2 + 3} \\ x^{5} \end{matrix}$

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 for Exponents to more than two factors.

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.


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:

$\begin{matrix} {(x^{2})}^{3} \\ x^{2 \cdot 3} \\ x^{6} \end{matrix}$

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

An example with numbers helps to verify this property.

$\begin{matrix} {(3^{2})}^{3} & \overset{?}{=} & 3^{2 \cdot 3} \\ {(9)}^{3} & \overset{?}{=} & 3^{6} \\ 729 & = & 729 ✓ \end{matrix}$

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. Can you find this 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 and $open parenthesis 2 x close parenthesis cubed$ 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.

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

Example 9

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

ⓐ	${(y^{3})}^{6} {(y^{5})}^{4}$
Use the Power Property.	$y^{18} \cdot y^{20}$
Add the exponents.	$y to the power 38$
ⓑ	${(−6 x^{4} y^{5})}^{2}$
Use the Product to a Power Property.	${(−6)}^{2} {(x^{4})}^{2} {(y^{5})}^{2}$
Use the Power Property.	$open parenthesis −6 close parenthesis squared$
Simplify.	$36 x^{8} y^{10}$

Example 10

Simplify: ⓐ $open parenthesis 5 m close parenthesis squared open parenthesis 3 m cubed close parenthesis$ ⓑ $open parenthesis 3 x squared y close parenthesis to the power 4 open parenthesis 2 x y squared close parenthesis cubed.$

ⓐ	$open parenthesis 5 m close parenthesis squared open parenthesis 3 m cubed close parenthesis$
Raise $5 m$ to the second power.	$5 squared m squared times 3 m cubed$
Simplify.	$25 m squared times 3 m cubed$
Use the Commutative Property.	$25 \cdot 3 \cdot m^{2} \cdot m^{3}$
Multiply the constants and add the exponents.	$75 m to the power 5$
ⓑ	$open parenthesis 3 x squared y close parenthesis to the power 4 open parenthesis 2 x y squared close parenthesis cubed$
Use the Product to a Power Property.	$(3^{4} x^{8} y^{4}) (2^{3} x^{3} y^{6})$
Simplify.	$(81 x^{8} y^{4}) (8 x^{3} y^{6})$
Use the Commutative Property.	$81 \cdot 8 \cdot x^{8} \cdot x^{3} \cdot y^{4} \cdot y^{6}$
Multiply the constants and add the exponents.	$648 x^{11} y^{10}$

Multiply Monomials

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

Key Concepts

Exponential Notation
Properties of Exponents
- If $a, b$ are real numbers and $m, n$ are whole numbers, then
  $\begin{matrix} Product Property & a^{m} \cdot a^{n} & = & a^{m + n} \\ Power Property & {(a^{m})}^{n} & = & a^{m \cdot n} \\ Product to a Power & {(a b)}^{m} & = & a^{m} b^{m} \end{matrix}$

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

ⓐ $3 to the power 5$ ⓑ $9 to the power 1$ ⓒ ${(\frac{1}{3})}^{2}$ ⓓ $open parenthesis 0.2 close parenthesis to the power 4$

ⓐ $10 to the power 4$ ⓑ $17 to the power 1$ ⓒ ${(\frac{2}{9})}^{2}$ ⓓ $open parenthesis 0.5 close parenthesis cubed$

ⓐ 10,000 ⓑ 17 ⓒ $\frac{4}{81}$ ⓓ 0.125

ⓐ $2 to the power 6$ ⓑ $14 to the power 1$ ⓒ ${(\frac{2}{5})}^{3}$ ⓓ $open parenthesis 0.7 close parenthesis squared$

ⓐ $8 cubed$ ⓑ $8 to the power 1$ ⓒ ${(\frac{3}{4})}^{3}$ ⓓ $open parenthesis 0.4 close parenthesis cubed$

ⓐ 512 ⓑ 8 ⓒ $\frac{27}{64}$
ⓓ 0.064

ⓐ $open parenthesis −6 close parenthesis to the power 4$ ⓑ $− 6 to the power 4$

ⓐ $open parenthesis −2 close parenthesis to the power 6$ ⓑ $− 2 to the power 6$

ⓐ 64 ⓑ $−64$

ⓐ $− {(\frac{1}{4})}^{4}$ ⓑ ${(- \frac{1}{4})}^{4}$

ⓐ $− {(\frac{2}{3})}^{2}$ ⓑ ${(- \frac{2}{3})}^{2}$

ⓐ $- \frac{4}{9}$ ⓑ $\frac{4}{9}$

ⓐ $− 0.5 squared$ ⓑ $open parenthesis −0.5 close parenthesis squared$

ⓐ $− 0.1 to the power 4$ ⓑ $open parenthesis −0.1 close parenthesis to the power 4$

ⓐ $−0.0001$ ⓑ 0.0001

Simplify Expressions Using the Product Property for Exponents

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

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

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

$x to the power 6$

$n^{19} \cdot n^{12}$

$q^{27} \cdot q^{15}$

$q to the power 42$

ⓐ $4^{5} \cdot 4^{9}$ ⓑ $8^{9} \cdot 8$

ⓐ $3^{10} \cdot 3^{6}$ ⓑ $5 \cdot 5^{4}$

ⓐ $3 to the power 16$ ⓑ $5 to the power 5$

ⓐ $y \cdot y^{3}$ ⓑ $z^{25} \cdot z^{8}$

ⓐ $w^{5} \cdot w^{}$ ⓑ $u^{41} \cdot u^{53}$

ⓐ $w to the power 6$ ⓑ $u to the power 94$

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

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

$y to the power 9$

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

$c^{5} \cdot c^{11} \cdot c^{2}$

$c to the power 18$

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

$n^{y} \cdot n^{2}$

$n^{y + 2}$

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

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

$x^{p + q}$

Simplify Expressions Using the Power Property for Exponents

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

ⓐ $open parenthesis m to the power 4 close parenthesis squared$ ⓑ $open parenthesis 10 cubed close parenthesis to the power 6$

ⓐ $open parenthesis b squared close parenthesis to the power 7$ ⓑ $open parenthesis 3 to the power 8 close parenthesis squared$

ⓐ $b to the power 14$ ⓑ $3 to the power 16$

ⓐ ${(y^{3})}^{x}$ ⓑ ${(5^{x})}^{y}$

ⓐ ${(x^{2})}^{y}$ ⓑ ${(7^{a})}^{b}$

ⓐ $x to the power 2 y$ ⓑ $7 to the power a b$

Simplify Expressions Using the Product to a Power Property

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

ⓐ $open parenthesis 6 a close parenthesis squared$ ⓑ $open parenthesis 3 x y close parenthesis squared$

ⓐ $open parenthesis 5 x close parenthesis squared$ ⓑ $open parenthesis 4 a b close parenthesis squared$

ⓐ $25 x squared$ ⓑ $16 a squared b squared$

ⓐ $open parenthesis −4 m close parenthesis cubed$ ⓑ $open parenthesis 5 a b close parenthesis cubed$

ⓐ $open parenthesis −7 n close parenthesis cubed$ ⓑ $open parenthesis 3 x y z close parenthesis to the power 4$

ⓐ $−343 n cubed$ ⓑ $81 x^{4} y^{4} z^{4}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

ⓐ ${(y^{2})}^{4} \cdot {(y^{3})}^{2}$ ⓑ $open parenthesis 10 a squared b close parenthesis cubed$

ⓐ ${(w^{4})}^{3} \cdot {(w^{5})}^{2}$ ⓑ ${(2 x y^{4})}^{5}$

ⓐ $w to the power 22$ ⓑ $32 x^{5} y^{20}$

ⓐ $open parenthesis −2 r cubed s squared close parenthesis to the power 4$ ⓑ ${(m^{5})}^{3} \cdot {(m^{9})}^{4}$

ⓐ $open parenthesis −10 q squared p to the power 4 close parenthesis cubed$ ⓑ ${(n^{3})}^{10} \cdot {(n^{5})}^{2}$

ⓐ $−1000 q^{6} p^{12}$ ⓑ $n to the power 40$

ⓐ $open parenthesis 3 x close parenthesis squared open parenthesis 5 x close parenthesis$ ⓑ $open parenthesis 5 t squared close parenthesis cubed open parenthesis 3 t close parenthesis squared$

ⓐ $open parenthesis 2 y close parenthesis cubed open parenthesis 6 y close parenthesis$ ⓑ ${(10 k^{4})}^{3} {(5 k^{6})}^{2}$

ⓐ $48 y to the power 4$ ⓑ $25,000 k to the power 24$

ⓐ $open parenthesis 5 a close parenthesis squared open parenthesis 2 a close parenthesis cubed$ ⓑ ${(\frac{1}{2} y^{2})}^{3} {(\frac{2}{3} y)}^{2}$

ⓐ $open parenthesis 4 b close parenthesis squared open parenthesis 3 b close parenthesis cubed$ ⓑ ${(\frac{1}{2} j^{2})}^{5} {(\frac{2}{5} j^{3})}^{2}$

ⓐ $432 b to the power 5$ ⓑ $\frac{1}{200} j^{16}$

ⓐ ${(\frac{2}{5} x^{2} y)}^{3}$ ⓑ ${(\frac{8}{9} x y^{4})}^{2}$

ⓐ $open parenthesis 2 r squared close parenthesis cubed open parenthesis 4 r close parenthesis squared$ ⓑ ${(3 x^{3})}^{3} {(x^{5})}^{4}$

ⓐ $128 r to the power 8$ ⓑ $27 x to the power 29$

ⓐ ${(m^{2} n)}^{2} {(2 m n^{5})}^{4}$ ⓑ ${(3 p q^{4})}^{2} {(6 p^{6} q)}^{2}$

Multiply Monomials

In the following exercises, multiply the monomials.

$(6 y^{7}) (−3 y^{4})$

$−18 y to the power 11$

$open parenthesis −10 x to the power 5 close parenthesis open parenthesis −3 x cubed close parenthesis$

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

$72 u to the power 7$

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

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

$4 f to the power 11$

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

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

$36 a^{5} b^{7}$

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

$(\frac{4}{7} r s^{2}) (14 r s^{3})$

$8 r squared s to the power 5$

$(\frac{5}{8} x^{3} y) (24 x^{5} y)$

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

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

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

Mixed Practice

In the following exercises, simplify each expression.

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

$x to the power 14$

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

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

$a to the power 36$

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

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

$8 m to the power 18$

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

$open parenthesis 10 x squared y close parenthesis cubed$

$1000 x to the power 6 y cubed$

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

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

$16 a^{12} b^{8}$

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

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

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

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

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

$1024 a to the power 10$

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

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

$25000 p to the power 24$

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

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

$x^{18} y^{18}$

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

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

$144 m^{8} n^{22}$

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

Everyday Math

Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is $10 squared$ , on the third round is $10 cubed$ , as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Round	Number of people
1	10
2	$10 squared$
3	$10 cubed$
…	…
6	?

$1,000,000$

Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $$ 35,000 open parenthesis 1.03 close parenthesis$ , after 2 years was $$ 35,000 open parenthesis 1.03 close parenthesis squared$ , after 3 years was $$ 35,000 open parenthesis 1.03 close parenthesis cubed$ , as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.

Year	Salary
1	$$ 35,000 open parenthesis 1.03 close parenthesis$
2	$$ 35,000 open parenthesis 1.03 close parenthesis squared$
3	$$ 35,000 open parenthesis 1.03 close parenthesis cubed$
…	…
10	?

Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $$ 1,000 open parenthesis 0.70 close parenthesis$ and the cost of the item during the second week would be $$ 1,000 open parenthesis 0.70 close parenthesis squared$ . Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Week	Cost
1	$$ 1,000 open parenthesis 0.70 close parenthesis$
2	$$ 1,000 open parenthesis 0.70 close parenthesis squared$
3
4	…
5	?

$168.07

Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for $20,000, the value at the end of the first year would be $$ 20,000 open parenthesis 0.90 close parenthesis$ and the value of the car after the end of the second year would be $$ 20,000 open parenthesis 0.90 close parenthesis squared$ . Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Year	Cost
1	$$ 20,000 open parenthesis 0.90 close parenthesis$
2	$$ 20,000 open parenthesis 0.90 close parenthesis squared$
3
…	…
8	?

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.

This is a table that has seven rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “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,” and “multiply monomials.” The rest of the cells are blank.

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

ⓐ	$open parenthesis −5 close parenthesis to the power 4$
Multiply four factors of $−5$ .	$open parenthesis −5 close parenthesis open parenthesis −5 close parenthesis open parenthesis −5 close parenthesis open parenthesis −5 close parenthesis$
Simplify.	$625$
ⓑ	$− 5 to the power 4$
Multiply four factors of 5.	$− (5 \cdot 5 \cdot 5 \cdot 5)$
Simplify.	$−625$

Product Property	$\begin{matrix} a^{m} \cdot a^{n} & = & a^{m + n} \end{matrix}$
Power Property	$\begin{matrix} {(a^{m})}^{n} & = & a^{m \cdot n} \end{matrix}$
Product to a Power	$\begin{matrix} {(a b)}^{m} & = & a^{m} b^{m} \end{matrix}$

	$open parenthesis 3 x squared close parenthesis open parenthesis −4 x cubed close parenthesis$
Use the Commutative Property to rearrange the terms.	$3 \cdot (−4) \cdot x^{2} \cdot x^{3}$
Multiply.	$−12 x to the power 5$

	$(\frac{5}{6} x^{3} y) (12 x y^{2})$
Use the Commutative Property to rearrange the terms.	$\frac{5}{6} \cdot 12 \cdot x^{3} \cdot x \cdot y \cdot y^{2}$
Multiply.	$10 x to the power 4 y cubed$


Rewrite, a = a¹.
Use the product property, a^m · aⁿ = a^m+n.
Simplify.


Notice, the bases are the same, so add the exponents.
Simplify.


Use Power of a Product Property, (ab)^m = a^mb^m.
Simplify.