6.2 Use Multiplication Properties of Exponents

Learning objectives

Learning objectives

• 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

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, ${2}^{4}$ means to multiply 2 by itself 4 times, so ${2}^{4}$ 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 6.16

Solution

Example 6.17

Solution

Notice the similarities and differences in Example 6.17ⓐ and Example 6.17ⓑ ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the $(-5)$ 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.

We write:

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.

Example 6.18

Solution

Example 6.19

Solution

1. ⓐ Use the product property, $a^{m}$ · $a^{n}$ = $a^{m+n}$. Simplify.

2. ⓑ Use the product property, $a^{m}$ · $a^{n}$ = $a^{m+n}$. Simplify.

Example 6.20

Solution

1. ⓐ Rewrite, a = $a^{1}$. Use the product property, $a^{m}$ · $a^{n}$ = $a^{m+n}$. Simplify.

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

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

Example 6.21

Solution

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.

We write:

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

An example with numbers helps to verify this property.

Example 6.22

Solution

ⓐ

ⓑ

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?

Notice that each factor was raised to the power and ${(2x)}^{3}$ is ${2}^{3}\cdot{x}^{3}$.

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:

Example 6.23

Solution

1. ⓐ Use Power of a Product Property, (ab)${}^{m}$ = $a^{m}$$b^{m}$. Simplify.

2. ⓑ Use Power of a Product Property, (ab)${}^{m}$ = $a^{m}$$b^{m}$. Simplify.

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 $m\,\mathrm{and}\,n$. Right now, we only use whole number exponents.

Example 6.24

Solution

Example 6.25

Solution

Multiply Monomials

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

Example 6.26

Solution

Example 6.27

Solution

Section 6.2 Exercises

Section 6.2 Exercises

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

Simplify Expressions Using the Product Property for Exponents

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

Simplify Expressions Using the Power Property for Exponents

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

Simplify Expressions Using the Product to a Power Property

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

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Multiply Monomials

In the following exercises, multiply the monomials.

Mixed Practice

In the following exercises, simplify each expression.

Everyday Math

Writing Exercises

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

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

1. Simplify: ⓐ ${6}^{3}$ ⓑ ${15}^{1}$ ⓒ ${(\frac{3}{7})}^{2}$ ⓓ ${(0.43)}^{2}.$

2. Simplify: ⓐ ${2}^{5}$ ⓑ ${21}^{1}$ ⓒ ${(\frac{2}{5})}^{3}$ ⓓ ${(0.218)}^{2}.$

3. Simplify: ⓐ ${(-3)}^{4}$ ⓑ $-{3}^{4}.$

4. Simplify: ⓐ ${(-13)}^{2}$ ⓑ $-{13}^{2}.$

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

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

2. Simplify: ⓐ $5\cdot{5}^{5}$ ⓑ ${4}^{9}\cdot{4}^{9}.$

3. Simplify: ⓐ ${7}^{6}\cdot{7}^{8}$ ⓑ $10\cdot{10}^{10}.$

4. Simplify: ⓐ ${p}^{5}\cdot{}p$ ⓑ ${y}^{14}\cdot{y}^{29}.$

5. Simplify: ⓐ $z\cdot{z}^{7}$ ⓑ ${b}^{15}\cdot{b}^{34}.$

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

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

3. Simplify: ⓐ ${({b}^{7})}^{5}$ ⓑ ${({5}^{4})}^{3}.$

4. Simplify: ⓐ ${({z}^{6})}^{9}$ ⓑ ${({3}^{7})}^{7}.$

5. Simplify: ⓐ ${(-12y)}^{2}$ ⓑ ${(2wx)}^{5}.$

1. Simplify: ⓐ ${(5wx)}^{3}$ ⓑ ${(-3y)}^{3}.$

2. Simplify: ⓐ ${({a}^{4})}^{5}{({a}^{7})}^{4}$ ⓑ ${(-2{c}^{4}{d}^{2})}^{3}.$

3. Simplify: ⓐ ${(-3{x}^{6}{y}^{7})}^{4}$ ⓑ ${({q}^{4})}^{5}{({q}^{3})}^{3}.$

4. Simplify: ⓐ ${(5n)}^{2}(3{n}^{10})$ ⓑ ${({c}^{4}{d}^{2})}^{5}{(3c{d}^{5})}^{4}.$

5. Simplify: ⓐ ${({a}^{3}{b}^{2})}^{6}{(4a{b}^{3})}^{4}$ ⓑ ${(2x)}^{3}(5{x}^{7}).$

1. Multiply: $(5{y}^{7})(-7{y}^{4}).$

2. Multiply: $(-6{b}^{4})(-9{b}^{5}).$

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

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

5. ⓐ ${10}^{4}$ ⓑ ${17}^{1}$ ⓒ ${(\frac{2}{9})}^{2}$ ⓓ ${(0.5)}^{3}$

1. ⓐ ${8}^{3}$ ⓑ ${8}^{1}$ ⓒ ${(\frac{3}{4})}^{3}$ ⓓ ${(0.4)}^{3}$

2. ⓐ ${(-6)}^{4}$ ⓑ $-{6}^{4}$

3. ⓐ ${(-2)}^{6}$ ⓑ $-{2}^{6}$

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

5. ⓐ $-{0.1}^{4}$ ⓑ ${(-0.1)}^{4}$

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

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

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

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

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

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

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

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

4. ⓐ ${({b}^{2})}^{7}$ ⓑ ${({3}^{8})}^{2}$

5. ⓐ ${({x}^{2})}^{y}$ ⓑ ${({7}^{a})}^{b}$

1. ⓐ ${(5x)}^{2}$ ⓑ ${(4ab)}^{2}$

2. ⓐ ${(-7n)}^{3}$ ⓑ ${(3xyz)}^{4}$

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

4. ⓐ ${(-10{q}^{2}{p}^{4})}^{3}$ ⓑ ${({n}^{3})}^{10}\cdot{({n}^{5})}^{2}$

5. ⓐ ${(2y)}^{3}(6y)$ ⓑ ${(10{k}^{4})}^{3}{(5{k}^{6})}^{2}$

1. ⓐ ${(4b)}^{2}{(3b)}^{3}$ ⓑ ${(\frac{1}{2}{j}^{2})}^{5}{(\frac{2}{5}{j}^{3})}^{2}$

2. ⓐ ${(2{r}^{2})}^{3}{(4r)}^{2}$ ⓑ ${(3{x}^{3})}^{3}{({x}^{5})}^{4}$

3. $(6{y}^{7})(-3{y}^{4})$

4. $(-8{u}^{6})(-9u)$

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

1. $(4{a}^{3}b)(9{a}^{2}{b}^{6})$

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

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

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

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

1. ${(2{m}^{6})}^{3}$

2. ${(10{x}^{2}y)}^{3}$

3. ${(-2{a}^{3}{b}^{2})}^{4}$

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

5. ${(8{a}^{3})}^{2}{(2a)}^{4}$

1. ${(10{p}^{4})}^{3}{(5{p}^{6})}^{2}$

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

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

4. 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 ${10}^{2}$, on the third round is ${10}^{3}$, 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.

5. 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(0.70)$ and the cost of the item during the second week would be

1,000(0.70)$ |

| 2 | $$1,000{(0.70)}^{2}$ |

| 3 | |

| 4 | … |

| 5 | ? |

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

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