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On this page you will find some material about Lesson 18. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 18:

Simplifying Radical Expressions

Addition and Subtraction of Radicals

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.

  • Recognize a radical expression in simplified form.
  • Use the multiplication property.
  • Identify like radical terms.
  • Simplify expressions with addition and subtraction of radicals.

Topic. This lesson covers

Section 6.3: Simplifying Radical Expressions, and

Section 6.4: Addition and Subtraction of Radicals.

WeBWorK. There are two WeBWorK assignments on today’s material:

SimplifyingRadicals

AddSubtractRadicals

Lesson Notes.

Video Lessons.

Video Lesson 18 – part 1 (based on Lesson 18 Notes – part 1)

Video Lesson 18 – part 2 (based on Lesson 18 Notes – part 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

Simplify ((-27)^{1/3})^2.

Show Answer 1

    \[((-27)^{1/3})^2 =(((-3)^3)^{1/3})^2 = (-3)^2=9\]

Warmup Question 2

Simplify -36^{-1/2}

Show Answer 2

-36^{-1/2}=-\dfrac{1}{36^{1/2}}=-\dfrac{1}{\sqrt{36}}=-\dfrac{1}{6}

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 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 Simplifying Radical Expressions & Addition and Subtraction of Radicals

Key Words. Radicals, radicand, index, simplified form, like radicals, addition/subtraction of radicals.

The goal of this lesson is to simplify radical expressions. What does this mean?

  • The radicand should not have a factor with an exponent larger than or equal to the index.
    • For example, \sqrt[4]{x^5} in not in simplified form.
  • There should be no fraction in the radicand.
    • For example, \sqrt{\dfrac{1}{16}} is not in simplified form.
  • There should be no radicals in the denominator of a fraction.
    • For example, \dfrac{1}{\sqrt{2}} is not in simplified form.

The last case will be dealt in ????.. For the second case, we can simplify the radical expression as

    \[\sqrt{\dfrac{1}{16}}=\sqrt{\left(\dfrac{1}{4}\right)^2}=\dfrac{1}{4}\]

For the first case, we need the following

Multiplication Property:

    \[\sqrt[n]{ab}=\sqrt[n] a\cdot\sqrt[n] b\]

Now we can simplify \sqrt[4]{x^5} as

    \[\sqrt[4]{x^5}= \sqrt[4]{xx^4} = \sqrt[4]{x^4}\sqrt[4]{x}=x\sqrt[4]{x}.\]

Video Lesson I

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 1

A video lesson on Simplifying Radical Expressions [9:45]

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 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

Simplify \sqrt{180}.

Show Answer 1

    \[\sqrt{180}=\sqrt{36\cdot 5} = \sqrt{36}\sqrt{5} = 6 \sqrt{5}\]

Try Question 2

Simplify \sqrt{27x^3y^7}.

Show Answer 2

    \[\sqrt{27x^3y^7}= \sqrt{(9x^2y^6)(3xy)}= \sqrt{9x^2y^6}\sqrt{3xy}\]

    \[= \sqrt{(3xy^3)^2}\sqrt{3xy}= 3xy^3\sqrt{3xy}\]

Try Question 3

Simplify 5x^2y\sqrt{3x^2y^7}.

Show Answer 3

    \[5x^2y\sqrt{3x^2y^7}= 5x^2y\sqrt{(x^2y^6)3y}= 5x^2y\sqrt{x^2y^6}\sqrt{3y}\]

    \[=5x^2y\sqrt{(xy^3)^2}\sqrt{3y} = 5x^2yxy^3\sqrt{3y} = 5x^3y^4\sqrt{3y}\]

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.

Adding and Subtracting Radicals

Like radicals are radical terms with the same index and radicand. We add and subtract radical expressions by factoring like radicals.

    \[-5\sqrt{a}+8\sqrt{a}=(-5+8)\sqrt{a} = 3\sqrt{a}\]

Video Lesson II

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 2

Video lesson on Adding and Subtracting Radicals [9:59]

A description of the video

In the video you will see the following radical expressions.

  • \sqrt 3+ 2
  • \sqrt{xy}+\sqrt 3
  • 2(\sqrt 3+3)
  • \dfrac{3\sqrt 5 +2 }{2}
  • (3\sqrt 5 +2)+(-2\sqrt 5+3)
  • (3\sqrt 5+2)-(-2\sqrt 5+3)
  • 3\sqrt 5 +2\sqrt 7
  • 3\sqrt{20}-2\sqrt{5}

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 4

Simplify -2\sqrt 5 - 3 \sqrt{50}+\sqrt{125}.

Show Answer 4

    \[-2\sqrt 5 - 3 \sqrt{50}+\sqrt{125} = -2\sqrt 5 - 3 \sqrt{25\cdot 2}+\sqrt{25\cdot 5}\]

    \[= -2\sqrt 5 - 3 \sqrt{25}\cdot \sqrt{2}+\sqrt{25}\cdot \sqrt{5}\]

    \[= -2\sqrt 5 - 3\cdot 5 \sqrt{2}+5 \sqrt{5} =& -2\sqrt 5 - 15 \sqrt 2 + 5\sqrt 5\]

    \[= -2\sqrt 5 + 5\sqrt 5- 15 \sqrt 2  = (-2+5)\sqrt 5 - 15 \sqrt 2 = 3\sqrt 5 - 15 \sqrt 2\]

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:

SimplifyingRadicals

AddSubtractRadicals

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 the goal in simplifying a radical expression? 
  • How can you recognize if a radical expression is simplified? 
  • Why do often need to simplify radical expressions before adding them?  Give an example.
  • What are like radicals? What is their role in adding and subtracting radicals?

\bigstar Simplify

(a) 3\sqrt{80}-5\sqrt{45}

(b) -4x^3y\sqrt{32x^6y^9}

Show Answer

(a)

    \[3\sqrt{80}-5\sqrt{45}= 3\sqrt{16\cdot 5}-5\sqrt{9\cdot 5}=3\cdot 4\sqrt{5}-5\cdot 3\sqrt 5\]

    \[=12\sqrt{5}-15\sqrt{5}=(12-15)\sqrt 5=-3\sqrt 5\]

(b)

    \[-4x^3y\sqrt{32x^6y^9}= -4x^3y\sqrt{(16x^6y^8)(2y)}=-4x^3y\sqrt{16x^6y^8}\sqrt{2y}\]

    \[=-4x^3y\sqrt{(4x^3y^4)^2}\sqrt{2y}=-4x^3y4x^3y^4\sqrt{2y}=-16x^6y^5\sqrt{2y}\]

Need more help?

Don’t wait too long to do the following.

  • Watch the additional video resources.
Additional video resources on Simplifying Radical Expressions
Additional video resources on Addition and Subtraction of Radicals
  • Talk to your instructor.
  • Form a study group.
  • Visit a tutor. For more information, check the tutoring page.