Lesson 4: Roots and Rational Exponents

Resources

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:

Lesson Notes.

Video Lesson.

Video Lesson 4 (based on Lesson 4 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.

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.

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

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.

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 Lesson 1 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.

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!

Need more help?

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