10.3: Use Multiplication Properties of Exponents (Part 2)

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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.

	(2x)³
What does this mean?	2x • 2x • 2x
We group the like factors together.	2 • 2 • 2 • x • x • x
How many factors of 2 and of x?	2³ • x³
Notice that each factor was raised to the power.	(2x)³is 2³ • x³
We write:	$$\begin{split} &(2x)^{3} \\ &2^{3} \cdot x^{3} \end{split}$$

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

Definition: Product to a Power Property of Exponents

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

\[(ab)^{m} = a^{m} b^{m} \tag{10.2.27}\]

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

An example with numbers helps to verify this property:

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

Example $\PageIndex{10}$:

Simplify: (−11x)².

Solution

Use the Power of a Product Property, (ab)^m = a^m b^m.	$$(-11)^{\textcolor{red}{2}} x^{\textcolor{red}{2}} \tag{10.2.28}$$
Simplify.	$$121x^{2} \tag{10.2.29}$$

Exercise $\PageIndex{19}$:

Simplify: (−14x)².

Answer: 196x²

Exercise $\PageIndex{20}$:

Simplify: (−12a)².

Answer: 144a²

Example $\PageIndex{11}$:

Simplify: (3xy)³.

Solution

Raise each factor to the third power.	$$3^{\textcolor{red}{3}} x^{\textcolor{red}{3}} y^{\textcolor{red}{3}} \tag{10.2.30}$$
Simplify.	$$27x^{3} y^{3} \tag{10.2.31}$$

Exercise $\PageIndex{21}$:

Simplify: (−4xy)⁴.

Answer: 256x⁴y⁴

Exercise $\PageIndex{22}$:

Simplify: (6xy)³.

Answer: 216x³y³

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.

Definition: Properties of Exponents

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

Product Property	a^m • aⁿ = a^{m + n}
Power Property	(a^m)ⁿ = a^{m • n}
Product to a Power Property	(ab)^m = a^mb^m

Example $\PageIndex{12}$:

Simplify: (x²)⁶(x⁵)⁴.

Solution

Use the Power Property.	x¹²• x²⁰
Add the exponents.	x³²

Exercise $\PageIndex{23}$:

Simplify: (x⁴)³(x⁷)⁴.

Answer: x⁴⁰

Exercise $\PageIndex{24}$:

Simplify: (y⁹)²(y⁸)³.

Answer: y⁴²

Example $\PageIndex{13}$:

Simplify: (−7x³y⁴)².

Solution

Take each factor to the second power.	(−7)²(x³)²(y⁴)²
Use the Power Property.	49x⁶y⁸

Exercise $\PageIndex{25}$:

Simplify: (−8x⁴y⁷)³.

Answer: -512x¹²y²¹

Exercise $\PageIndex{26}$:

Simplify: (−3a⁵b⁶)⁴.

Answer: 81a²⁰b²⁴

Example $\PageIndex{14}$:

Simplify: (6n)²(4n³).

Solution

Raise 6n to the second power.	6²n² • 4n³
Simplify.	36n² • 4n³
Use the Commutative Property.	36 • 4 • n² • n³
Multiply the constants and add the exponents.	144n⁵

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.

Exercise $\PageIndex{27}$:

Simplify: (7n)² (2n¹²).

Answer: 98n¹⁴

Exercise $\PageIndex{28}$:

Simplify: (4m)²(3m³).

Answer: 48m⁵

Example $\PageIndex{15}$:

Simplify: (3p²q)⁴(2pq²)³.

Solution

Use the Power of a Product Property.	3⁴(p²)⁴q⁴ • 2³p³(q²)³
Use the Power Property.	81p⁸q⁴ • 8p³q⁶
Use the Commutative Property.	81 • 8 • p⁸ • p³ • q⁴ • q⁶
Multiply the constants and add the exponents for each variable.	648p¹¹q¹⁰

Exercise $\PageIndex{29}$:

Simplify: (u³v²)⁵(4uv⁴)³.

Answer: 64u¹⁸v²²

Exercise $\PageIndex{30}$:

Simplify: (5x²y³)²(3xy⁴)³.

Answer: 675x⁷y¹⁸

Multiply Monomials

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

Example $\PageIndex{16}$:

Multiply: (4x²)(−5x³).

Solution

Use the Commutative Property to rearrange the factors.	4 • (−5) • x² • x³
Multiply.	−20x⁵

Exercise $\PageIndex{31}$:

Multiply: (7x⁷)(−8x⁴).

Answer: -56x¹¹

Exercise $\PageIndex{32}$:

Multiply: (−9y⁴)(−6y⁵).

Answer: 54y⁹

Example $\PageIndex{17}$:

Multiply: $\left(\dfrac{3}{4} c^{3} d\right)$(12cd²).

Solution

Use the Commutative Property to rearrange the factors.	$\dfrac{3}{4}$ • 12 • c³ • c • d • d²
Multiply.	9c⁴d³

Exercise $\PageIndex{33}$:

Multiply: $\left(\dfrac{4}{5} m^{4} n^{3} d\right)$(15mn³).

Answer: 12m⁵n⁶

Exercise $\PageIndex{34}$:

Multiply: $\left(\dfrac{2}{3} p^{5} q d\right)$(18p⁶q⁷).

Answer: 12p¹¹q⁸

ACCESS ADDITIONAL ONLINE RESOURCES

Exponent Properties

Exponent Properties 2

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

4⁵
10³
$\left(\dfrac{1}{2}\right)^{2}$
$\left(\dfrac{3}{5}\right)^{2}$
(0.2)³
(0.4)³
(−5)⁴
(−3)⁵
−5⁴
−3⁵
−10⁴
−2⁶
$- \left(\dfrac{2}{3}\right)^{3}$
$- \left(\dfrac{1}{4}\right)^{4}$
−0.5²
−0.1⁴

Simplify Expressions Using the Product Property of Exponents

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

x³ • x⁶
m⁴ • m²
a • a⁴
y¹²• y
3⁵ • 3⁹
5¹⁰• 5⁶
z • z² • z³
a • a³ • a⁵
x^a • x²
y^p • y³
y^a • y^b
x^p • x^q

Simplify Expressions Using the Power Property of Exponents

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

(u⁴)²
(x²)⁷
(y⁵)⁴
(a³)²
(10²)⁶
(2⁸)³
(x¹⁵)⁶
(y¹²)⁸
(x²)^y
(y³)^x
(5^x)^y
(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.

(5a)²
(7x)²
(−6m)³
(−9n)³
(4rs)²
(5ab)³
(4xyz)⁴
(−5abc)³

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

(x²)⁴ • (x³)²
(y⁴)³ • (y⁵)²
(a²)⁶ • (a³)⁸
(b⁷)⁵ • (b²)⁶
(3x)²(5x)
(2y)³ (6y)
(5a)²(2a)³
(4b)²(3b)³
(2m⁶)³
(3y²)⁴
(10x²y)³
(2mn⁴)⁵
(−2a³b²)⁴
(−10u²v⁴)³
$\left(\dfrac{2}{3} x^{2} y \right)^{3}$
$\left(\dfrac{7}{9} p q^{4} \right)^{2}$
(8a³)²(2a)⁴
(5r²)³(3r)²
(10p⁴)³(5p⁶)²
(4x³)³(2x⁵)⁴
$\left(\dfrac{1}{2} x^{2} y^{3} \right)^{4}$ (4x⁵y³)²
$\left(\dfrac{1}{3} m^{3} n^{2} \right)^{4}$ (9m⁸n³)²
(3m²n)²(2mn⁵)⁴
(2pq⁴)³(5p⁶q)²

Multiply Monomials

In the following exercises, multiply the following monomials.

(12x²)(−5x⁴)
(−10y³)(7y²)
(−8u⁶)(−9u)
(−6c⁴)(−12c)
$\left(\dfrac{1}{5} r^{8} \right)$ (20r³)
$\left(\dfrac{1}{4} a^{5} \right)$ (36a²)
(4a³b)(9a²b⁶)
(6m⁴n³)(7mn⁵)
$\left(\dfrac{4}{7} x y^{2} \right)$ (14xy³)
$\left(\dfrac{5}{8} u^{3} v \right)^{3}$ (24u⁵v)
$\left(\dfrac{2}{3} x^{2} y \right) \left(\dfrac{3}{4} x y^{2} \right)$
$\left(\dfrac{3}{5} m^{3} n^{2} \right) \left(\dfrac{5}{9} m^{2} n^{3} \right)$

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 6², on the third round is 6³, 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	6²
3	6³
…	…
8	?

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(1.05), after 2 years was $40,000(1.05)², after 3 years was $40,000(1.05)³, 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(1.05)
2	$40,000(1.05)²
3	$40,000(1.05)³
…	…
10	?

Writing Exercises

Use the Product Property for Exponents to explain why x • x = x².
Explain why −5³ = (−5)³ but −5⁴ ≠ (−5)⁴.
Jorge thinks $\left(\dfrac{1}{2}\right)^{2}$ is 1. What is wrong with his reasoning?
Explain why x³• x⁵ is x⁸, and not x¹⁵.

Self Check

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

$CNX_BMath_Figure_AppB_060.jpg$

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

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."

Use the Commutative Property to rearrange the factors.	\(\dfrac{3}{4}\) • 12 • c³ • c • d • d²
Multiply.	9c⁴d³