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Lesson 16: Properties of Integer Exponents

### Resources

In this section you will find some important information about the specific resources related to this lesson:

• the learning outcomes,
• the section in the textbook,
• the WeBWorK homework sets,
• a link to the pdf of the lesson notes,
• a link to a video lesson.

Learning Outcomes.

• Simplify expressions with exponents.
• Apply the properties to rewrite the expression with positive exponents only.

Topic. This lesson covers

Section 4.1: Properties of Integer Exponents.

WeBWorK. There is one WeBWorK assignment on today’s material:

IntegerExponents

Lesson Notes.

Video Lesson.

Video Lesson 16 (based on Lesson 16 Notes)

### Warmup Questions

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

#### Warmup Question 1

Find $4\cdot 3$ and $4^3$.

$$4\cdot 3=\underbrace{4+ 4+ 4}_{3\text{ counts the number of 4’s being added}} = 12$$

$$4^3=\underbrace{4\cdot 4\cdot 4}_{3\text{ counts the number of 4’s being multiplied}} = 64$$

### Review

If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.

Need a review? Check ???.

### Quick Intro

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to Integer Exponents

Key Words. Exponentiation, base, exponent, power.

In the expression $b^n$, $b$ is called the base and $n$ is called the exponent or power. The operation is called exponentiation.

$\bigstar$ When $n$ is positive, the exponent $n$ is the number of $b$’s being multiplied.

$$b^n=\underbrace{b\cdot b \cdots b}_{n\text{ times}}$$

In the exponential expression $2^5$, the base is $2$ and the exponent is $5$.

$$2^5 =\underbrace{2\cdot 2 \cdot 2 \cdot 2 \cdot 2}_{5\text{ times}} = 32$$

What if the exponent $n$ is zero? Then $b^0=1$ for $b\neq 0$.

What if $n$ is negative? $b^{-n} = \dfrac{1}{b^n}$ for $b\neq 0$.

$\bigstar$ Properties:

1. $b^mb^n = b^{m+n}$
2. $\dfrac{b^m}{b^n} = b^{m-n}$
3. $(b^m)^n = b^{mn}$
4. $b^nc^n = (bc)^n$
5. $\dfrac{b^n}{c^n}= \left(\dfrac{b}{c}\right)^n$

### Video Lesson

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

Video Lesson

A description of the video

In the video you will see how to simplify:

• $2^3 2^4$
• $(4^2)^3$
• $2^35^3$
• $\dfrac{5^3}{4^3}$
• $(3x^5y^3)^2(2x^6y)^3$
• $\dfrac{(3x^5y)^2}{(9xy^2)^2}$
• $\left(\dfrac{2}{3}\right)^{-3}$
• $\dfrac{x^{-3}y}{x^2y^{-2}}$
• $3\left(\dfrac{2x^{-2}y^3z^{-5}}{3xy^2z^{-2}}\right)^2\left(\dfrac{-x^{2}y^{-2}}{zy}\right)^{-3}$

### Try Questions

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

#### Try Question 1

Find $5^3$.

$$5^3 = 5\cdot 5 \cdot 5 = 125$$

#### Try Question 2

Simplify $$(-2x^5y^2)^3(3x^6y^5)^2.$$

$$(-2x^5y^2)^3(3x^6y^5)^2= (-2)^3(x^5)^3(y^2)^3(3)^2(x^6)^2(y^5)^2$$

$$=-8x^{15}y^69x^{12}y^{10}$$

$$= (-8\cdot 9)(x^{15}x^{12})(y^6y^{10})$$

$$=-72x^{27}y^{16}$$

#### Try Question 3

Use Property (5) to simplify

$$\dfrac{(3x^5y)^2}{(9xy^2)^2}.$$

$$\dfrac{(3x^5y)^2}{(9xy^2)^2}= \left(\dfrac{3x^5y}{9xy^2}\right)^2}$$

$$=\left(\dfrac{x^4}{3y}\right)^2} =\dfrac{(x^4)^2}{(3y)^2}$$

$$=\dfrac{x^8}{9y^2}$$

### WeBWorK

You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.

WeBWorK

It is time to do the homework on WeBWork:

IntegerExponents

When you are done, come back to this page for the Exit Questions.

### Exit Questions

After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!

#### Exit Questions

• How is the negative exponent related to reciprocals? Give an example.
•  How are positive and negative exponents used in science to express large or small numbers?
• What is the purpose in writing numbers this way?

$\bigstar$ (1) Simplify $$\left(\dfrac{3a^{-3}}{b^{-5}}\right)^{-3}.$$ Write your final answer with positive exponents only.

$\bigstar$ (2) Simplify $$\dfrac{36x^5y^{10}}{70x^{15}y^5}.$$ Write your final answer with positive exponents only.

$\bigstar$ (1)

$$\left(\dfrac{3a^{-3}}{b^{-5}}\right)^{-3}= \dfrac{(3a^{-3})^{-3}}{(b^{-5})^{-3}}$$

$$=\dfrac{3^{-3}(a^{-3})^{-3}}{b^{15}}$$

$$=\dfrac{a^9}{3^3b^{15}}$$

$$=\dfrac{a^9}{27b^{15}}$$

$\bigstar$ (2)

$$\dfrac{36x^5y^{10}}{70x^{15}y^5}= \dfrac{2\cdot 18y^{10-5}}{2\cdot 35x^{15-5}}$$

$$=\dfrac{18y^5}{35x^{10}}$$

### Need more help?

Don’t wait too long to do the following.

• Watch the additional video resources.