Hi Everyone!

On this page you will find some material about Lesson 17. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

**Lesson 17: Roots and Rational Exponents**

**Learning Outcomes.**

- Simplify a radical expression.
- Identify the index and radicand.
- Simplify expressions with rational exponents.
- Use the properties of exponents.
- Convert a radical expression into an expression with rational exponents.
- Convert an expression with rational exponents into a radical expression.

**Topic**. This lesson covers

Section 6.1: Definition of an $n^{\mathrm{th}}$ Root, and

Section 6.2: Rational Exponents.

**WeBWorK.** There are three WeBWorK assignments on today’s material:

HigherRoots

HigherRoots-Algebraic

RationalExponents

**Lesson Notes.**

**Video Lesson.**

Video Lesson 17 (based on Lesson 17 Notes – parts 1 and 2)

### 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

What is $\sqrt{16}$?

#### Show Answer 1

$4$

#### Warmup Question 2

Simplify $\dfrac{(a^{-2}b^3)^{4}}{a}$. Write your answer with positive exponents only.

#### Show Answer 2

\begin{align*}&\dfrac{(a^{-2}b^3)^{4}}{a} \\=& \dfrac{a^{-8}b^{12}}{a}\\ =& a^{-9}b^{12}\\=&\dfrac{b^{12}}{a^9}\end{align*}

### 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 Lesson 16.

### Quick Intro I

*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 Radical Expressions**

**Key Words.** Radical expression, radical symbol, radicand, index, rational exponent.

The **radical expression** $\sqrt[a]{b}$ is the number (non-negative if $b$ is even) whose $b^{\text{th}}$ power is $a$. That is $(\sqrt[a]{b})^b=a$. We say that

$\bigstar$ $\sqrt$ is the **radical symbol**,

$\bigstar$ $a$ is the **radicand**,

$\bigstar$ $b$ is the **index**.

For example,

$$\sqrt[2]{25}=\sqrt{25}=5$$

since $5^2=25$ and $5\geq 0$. Another example:

$$\sqrt[3]{-27}=-3$$

since $(-3)^3=-27$.

### Try Questions I

*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 $$\sqrt[5]{-32}.$$

#### Show Answer 1

$\sqrt[5]{-32} =-2$ since $(-2)^5=-32$

### Quick Intro II

*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.*

An expression with a **rational exponent** can be written as a radical expression:

$$a^{1/b}=\sqrt[b]{a}.$$

For example, $64^{1/3} =\sqrt[3]{64} =4$.

All properties listed in ??? for integer exponents can be extended to rational exponents $m$ and $n$:

- $a^ma^n=a^{m+n}$
- $\dfrac{a^m}{a^n}=a^{m-n}$
- $(a^m)^n=a^{mn}$
- $(ab)^m = a^mb^m$
- $\left(\dfrac{a}{b}\right)^m = \dfrac{a^m}{b^m}$
- $a^{-m}=\dfrac{1}{a^m}$

In general,

$$a^{m/n}=(a^{1/n})^m=({\sqrt[n]{a})^m$$

or

$$a^{m/n}=(a^{m})^{1/n}=\sqrt[n]{a^m}.$$

### 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 the following examples:

- $\sqrt[3]{27x^3}$
- $\sqrt{x^9}$
- $\sqrt[3]{x^9}$
- $(3\sqrt{3x^2y})(\sqrt{15x^2y})$
- $\sqrt{150}$
- $\sqrt{125}$
- $\sqrt{40}$

### Try Questions II

*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 2

Simplify $$121^{1/2}+36^{1/2}.$$

#### Show Answer 2

\begin{align*}&121^{1/2}+36^{1/2} \\=& \sqrt{121}+\sqrt{36}\\ =& 11 + 6\\ =& 17\end{align*}

#### Try Question 3

Find $$(8x^6)^{1/3}.$$

#### Show Answer 3

\begin{align*}& (8x^6)^{1/3}\\=& \sqrt[3]{8x^6}\\ =& \sqrt[3]{(2x^2)^3}\\=&2x^2\end{align*}

#### Try Question 4

Simplify $$\left(p^{1/3} p^{1/4}\right)^{12}.$$

#### Show Answer 4

\begin{align*}& \left(p^{\frac{1}{3}}p^{\frac{1}{4}}\right)^{12} \\=&\left(p^{\frac{1}{3}+\frac{1}{4}}\right)^{12} \\=&\left(p^{\frac{7}{12}}\right)^{12} \\=& p^{\frac{7}{12}\cdot 12}\\ =& p^7 \end{align*}

#### Try Question 5

Simplify $$\sqrt[3]{-8x^3}.$$

#### Show Answer 5

\begin{align*}&\sqrt[3]{-8x^3}\\=& \sqrt[3]{(-2x)^3}\\=& -2x\end{align*}

#### Try Question 6

Simplify $$5x^2y\sqrt{16x^2y^8}.$$

#### Show Answer 6

\begin{align*}&5x^2y\sqrt{16x^2y^8} \\=& 5x^2y\sqrt{(4xy^4)^2}\\ =&5x^2y4xy^4 \\=& 20x^3y^5\end{align*}

### 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:

HigherRoots

HigherRoots-Algebraic

RationalExponents

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

- What is a square root?
- How is the square root related to rational exponents?
- What about a cube root?
- What happens if we square a square root?
- What is the index? Radicand?

$\bigstar$ Simplify

(a) $\sqrt{81a^4b^6}$

(b) $(-27b^9)^{1/3}$

#### Show Answer

(a) \begin{align*}&\sqrt{81a^4b^6} \\=& \sqrt{(9a^2b^3)^2}\\ =& 9a^2b^3\end{align*}

(b) \begin{align*}&(-27b^9)^{1/3}\\=& \sqrt[3]{-27b^9}\\=&\sqrt[3]{(-3b^3)^3} \\=& -3b^3\end{align*}

### Need more help?

*Don’t wait too long to do the following.*

- Watch the additional video resources.

- Talk to your instructor.
- Form a study group.
- Visit a tutor. For more information, check the tutoring page.