10.2: Use Multiplication Properties of Exponents (Part 1)

Learning Objectives

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

be prepared!

Before you get started, take this readiness quiz.

1. Simplify: $\dfrac{3}{4} \cdot \dfrac{3}{4}$. If you missed the problem, review Example 4.3.7.
2. Simplify: (−2)(−2)(−2). If you missed the problem, review Example 3.7.6.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 2⁴ means to multiply four factors of 2, so 2⁴ means 2 • 2 • 2 • 2. This format is known as exponential notation.

Definition: Exponential Notation

This is read a to the m^th power.

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

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

Example $\PageIndex{1}$:

Simplify: (a) 5³ (b) 9¹

Solution

(a) 5³

Multiply 3 factors of 5.	5 • 5 • 5
Simplify.	125

(b) 9¹

Multiply 1 factor of 9.

9

Exercise $\PageIndex{1}$:

Simplify: (a) 4³ (b) 11¹

Answer a

64

Answer b

11

Exercise $\PageIndex{2}$:

Simplify: (a) 3⁴ (b) 21¹

Answer a

81

Answer b

21

Example $\PageIndex{2}$:

Simplify: (a) $\left(\dfrac{7}{8}\right)^{2}$ (b) (0.74)²

Solution

(a) $\left(\dfrac{7}{8}\right)^{2}$

Multiply two factors.	$$\left(\dfrac{7}{8}\right) \left(\dfrac{7}{8}\right)$$
Simplify.	$$\dfrac{49}{64}$$

(b) (0.74)²

Multiply two factors.	(0.74)(0.74)
Simplify.	0.5476

Exercise $\PageIndex{3}$:

Simplify: (a) $\left(\dfrac{5}{8}\right)^{2}$ (b) (0.67)²

Answer a

$\frac{25}{64}$

Answer b

0.4489

Exercise $\PageIndex{4}$:

Simplify: (a) $\left(\dfrac{2}{5}\right)^{3}$ (b) (0.127)²

Answer a

$\frac{8}{125}$

Answer b

0.016129

Example $\PageIndex{3}$:

Simplify: (a) (−3)⁴ (b) −3⁴

Solution

(a) (−3)⁴

Multiply four factors of −3.	(−3)(−3)(−3)(−3)
Simplify.	81

(b) −3⁴

Multiply two factors.	−(3 • 3 • 3 • 3)
Simplify.	−81

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

Exercise $\PageIndex{5}$:

Simplify: (a) (−2)⁴ (b) −2⁴

Answer a

16

Answer b

-16

Exercise $\PageIndex{6}$:

Simplify: (a) (−8)² (b) −8²

Answer a

64

Answer b

-64

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.

	$$x^{2} \cdot x^{2}$$
What does this mean? How many factors altogether?
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:	$$\begin{split} &x^{2} \cdot x^{3} \\ &x^{2+3} \\ &x^{5} \end{split}$$

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

Definition: 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.

An example with numbers helps to verify this property.

$$\begin{split} 2^{2} \cdot 2^{3} &\stackrel{?}{=} 2^{2+3} \\ 4 \cdot 8 &\stackrel{?}{=} 2^{5} \\ 32 &= 32\; \checkmark \end{split}$$

Example $\PageIndex{4}$:

Simplify: x⁵ • x⁷.

Solution

Use the product property, a^m • a^n = a^m + n.	$$x^{\textcolor{red}{5+7}}$$
Simplify.	$$x^{12}$$

Exercise $\PageIndex{7}$:

Simplify: x⁷ • x⁸.

Answer

x¹⁵

Exercise $\PageIndex{8}$:

Simplify: x⁵ • x¹¹.

Answer

x¹⁶

Example $\PageIndex{5}$:

Simplify: b⁴ • b.

Solution

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

Exercise $\PageIndex{9}$:

Simplify: p⁹ • p.

Answer

p¹⁰

Exercise $\PageIndex{10}$:

Simplify: m • m⁷.

Answer

m⁸

Example $\PageIndex{6}$:

Simplify: 2⁷ • 2⁹.

Solution

Use the product property, a^m • a^n = a^m + n.	$$2^{\textcolor{red}{7+9}}$$
Simplify.	$$2^{16}$$

Exercise $\PageIndex{11}$:

Simplify: 6 • 6⁹.

Answer

6¹⁰

Exercise $\PageIndex{12}$:

Simplify: 9⁶ • 9⁹.

Answer

9¹⁵

Example $\PageIndex{7}$:

Simplify: y¹⁷ • y²³.

Solution

Notice, the bases are the same, so add the exponents.	$$y^{\textcolor{red}{17+23}}$$
Simplify.	$$y^{40}$$

Exercise $\PageIndex{13}$:

Simplify: y²⁴ • y¹⁹.

Answer

y⁴³

Exercise $\PageIndex{14}$:

Simplify: z¹⁵ • z²⁴.

Answer

z³⁹

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

Example $\PageIndex{8}$:

Simplify: x³ • x⁴ • x².

Solution

Add the exponents, since the bases are the same.	$$x^{\textcolor{red}{3+4+2}}$$
Simplify.	$$x^{9}$$

Exercise $\PageIndex{15}$:

Simplify: x⁷ • x⁵ • x⁹.

Answer

x²¹

Exercise $\PageIndex{16}$:

Simplify: y³ • y⁸ • y⁴.

Answer

y¹⁵

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.

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

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

Definition: Power Property of Exponents

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

$$(a^{m})^{n} = a^{m \cdot n}$$

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

An example with numbers helps to verify this property.

$$\begin{split} (5^{2})^{3} &\stackrel{?}{=} 5^{2 \cdot 3} \\ (25)^{3} &\stackrel{?}{=} 5^{6} \\ 15,625 &= 15,625\; \checkmark \end{split}$$

Example $\PageIndex{9}$:

Simplify: (a) (x⁵)⁷ (b) (3⁶)⁸

Solution

(a) (x⁵)⁷

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

(b) (3⁶)⁸

Use the Power Property, (a^m)^n = a^m • n.	$$3^{\textcolor{red}{6 \cdot 8}}$$
Simplify.	$$x^{48}$$

Exercise $\PageIndex{17}$:

Simplify: (a) (x⁷)⁴ (b) (7⁴)⁸

Answer a

x²⁸

Answer b

7³²

Exercise $\PageIndex{18}$:

Simplify: (a) (x⁶)⁹ (b) (8⁶)⁷

Answer a

y⁵⁴

Answer b

8⁴²

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."