- Toggle navigation
Mathematics Department Course Hub
- Project Profile
- Course Outline
- Official Textbooks
- Video resources
- Faculty Announcements
- Course Coordination
- WeBWorK – Faculty Resources
- Training and Support
- STEM Applications
- Lesson 7: Division of Radicals and Rationalization
Hi Everyone!
On this page you will find some material about Lesson 7. Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Table of Contents
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.
- Divide radical expressions.
- Find the conjugate of a radical expression.
- Rationalize an expression with a radical term in the denominator.
- Simplify radical expressions.
Topic . This lesson covers Section 6.6: Division of Radicals and Rationalization.
WeBWorK . There is one WeBWorK assignment on today’s material:
RationalizeDenominators
Lesson Notes.
Video Lesson.
Video Lesson 7 (based on Lesson 7 Notes)
![](http://rss3.fun/777/templates/cheerup/res/banner1.gif)
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
$$(2\sqrt{5a})(-4\sqrt{5a}).$$
Show Answer 1
$$(2\sqrt{5a})(-4\sqrt{5a})= -8\sqrt{(5a)^2}=-8(5a)=-40a$$
Warmup Question 2
$$(\sqrt 3-6)(\sqrt 3+6).$$
Show Answer 2
$$(\sqrt 3-6)(\sqrt 3+6)= (\sqrt 3)^2-6^2 = 3 – 36 = -33$$
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 6 .
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 Division of Radicals and Rationalization
Key Words. Radicals, division of radicals, simplified form, rationalization, conjugate.
When dividing radical terms, the following property can be very helpful.
Division Property
$$\dfrac{\sqrt[n]a}{\sqrt[n]b}=\sqrt[n]{\dfrac{a}{b}}$$
If not, it may be necessary to rationalize the denominator. On Lesson 5 we listed three conditions for a radical expression to be in simplified form. The third one is:
There should be no radicals in the denominator of a fraction.
$\bigstar$ We gave $\dfrac{1}{\sqrt 2}$ as an example that fails this condition. To simplify it, we multiply both the numerator and the denominator by $\sqrt 2$.
$$\underbrace{\dfrac{1}{\sqrt 2}}_{\text{radical in the denominator}}=\dfrac{1}{\sqrt 2}\cdot\dfrac{\sqrt 2}{\sqrt 2}=\dfrac{1\cdot\sqrt 2}{\sqrt 2\cdot\sqrt 2}=\underbrace{\dfrac{\sqrt 2}{2}}_{\text{no radical in the denominator}}$$
$\bigstar$ The process of removing a radical from the denominator is called rationalization .
$\bigstar$ The key idea was to multiply the original denominator by another copy of it, since squaring eliminates the radical.
$$(\sqrt a)(\sqrt a) = (\sqrt a)^2=a.$$
$\bigstar$ But what if we have $\dfrac{1}{\sqrt 3 -6}$? Squaring $\sqrt{3}-6$ will not help (try it!). In the Warmup Question #2 we saw that
$$(\sqrt 3 -6)(\sqrt 3 +6)=-33$$
results in a number free of radical. So
$$\underbrace{\dfrac{1}{\sqrt 3 -6}}_{\text{radical in the denominator}}=\dfrac{1}{\sqrt 3 -6}\cdot\dfrac{\sqrt 3 +6}{\sqrt 3 +6}$$
$$=\dfrac{1\cdot(\sqrt 3 +6)}{(\sqrt 3 -6)\cdot(\sqrt 3 +6)}=\underbrace{-\dfrac{\sqrt 3 +6}{33}}_{\text{no radical in the denominator}}$$
$\bigstar$ This happens because the above product is a difference of squares
$$(a-b)(a+b)=a^2-b^2,$$
and squaring a single radical eliminates the radical.
$\bigstar$ We say that $a-b$ and $a+b$ are conjugates . So if the denominator is $\sqrt 3 -6$, we rationalize it by multiplying the numerator and the denominator by its conjugate $\sqrt 3+6$.
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!
A description of the video
In the video you will see the following radical expressions.
- $\dfrac{3\sqrt x+1}{2\sqrt x}$
- $\dfrac{3\sqrt 5 + 1}{2\sqrt 7 -3}$
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
Simplify $$\dfrac{\sqrt{15}}{5\sqrt{20}}.$$
$$\dfrac{\sqrt{15}}{5\sqrt{20}} = \dfrac{\sqrt{15}}{5\sqrt{20}}\cdot \dfrac{\sqrt{20}}{\sqrt{20}}$$
$$= \dfrac{\sqrt{15}\sqrt{20}}{5\cdot 20} = \dfrac{\sqrt{15} \cdot 2\sqrt{5}}{5\cdot 20}$$
$$= \dfrac{\sqrt {5\cdot 15}}{5\cdot 10} = \dfrac{5\sqrt {3}}{50} = \dfrac{\sqrt 3}{10}$$
Try Question 2
Simplify $$\dfrac{3}{4+2\sqrt 5}.$$
$$\dfrac{3}{4+2\sqrt 5}=\dfrac{3}{4+2\sqrt 5}\cdot \dfrac{4-2\sqrt 5}{4-2\sqrt 5}$$
$$= \dfrac{3(4-2\sqrt 5)}{(4+2\sqrt 5)(4-2\sqrt 5)}= \dfrac{12-6\sqrt 5}{16-20}$$
$$=\dfrac{12-6\sqrt 5}{-4} = \dfrac{-6+3\sqrt 5}{2} $$
Try Question 3
Rationalize and simplify
$$\dfrac{1+3\sqrt 2}{1-\sqrt 2}+\sqrt 2.$$
Show Answer 3
$$\dfrac{1+3\sqrt 2}{1-\sqrt 2}+\sqrt 2= \dfrac{1+3\sqrt 2}{1-\sqrt 2}\cdot\dfrac{1+\sqrt 2}{1+\sqrt 2}+\sqrt 2$$
$$=\dfrac{(1+3\sqrt 2)(1+\sqrt 2)}{(1-\sqrt 2)(1+\sqrt 2)} +\sqrt 2 = \dfrac{1+\sqrt 2+3\sqrt 2 +3\cdot 2}{1-2}+\sqrt 2 $$
$$=\dfrac{7+4\sqrt 2}{-1}+\sqrt 2 = -7-4\sqrt 2+\sqrt 2 = -7-3\sqrt 2$$
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.
It is time to do the homework on WeBWork:
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!
- What is the goal in rationalizing the denominator?
- Why does the ‘conjugate’ play a role in accomplishing this?
$\bigstar$ Simplify
(a) $\dfrac{3-3\sqrt{3a}}{4\sqrt{8a}}$
(b) $\dfrac{\sqrt{5}+3}{4-\sqrt 5}$
![](http://rss3.fun/777/templates/cheerup/res/banner1.gif)
Show Answer
(a) $$\dfrac{3-3\sqrt{3a}}{4\sqrt{8a}}=\dfrac{3-3\sqrt{3a}}{4\cdot 2 \sqrt{2a}} $$
$$= \dfrac{3-3\sqrt{3a}}{8\sqrt{2a}}\cdot\dfrac{\sqrt{2a}}{\sqrt{2a}} =\dfrac{(3-3\sqrt{3a}) \sqrt{2a}}{8\cdot 2a} $$
$$= \dfrac{3\sqrt{2a}-3\sqrt{6a^2}}{16a} =\dfrac{3\sqrt{2a}-3a\sqrt 6}{16a}$$
(b) $$\dfrac{\sqrt{5}+3}{4-\sqrt 5}= \dfrac{\sqrt{5}+3}{4-\sqrt 5}\cdot \dfrac{4+\sqrt 5}{4+\sqrt 5} $$
$$= \dfrac{(\sqrt{5}+3)(4+\sqrt 5)}{(4-\sqrt 5)(4-\sqrt 5)}=\dfrac{4\sqrt{5}+5+12+3\sqrt 5}{16-5}$$
$$ =\dfrac{7\sqrt 5+17}{11}$$
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 .
Lessons Menu
- Lesson 1: Properties of Integer Exponents & Addition and Subtraction of Rational Expressions
- Lesson 2: Complex Fractions
- Lesson 3: Solving Rational Equations
- Lesson 4: Roots and Rational Exponents
- Lesson 5: Simplifying Radical Expressions & Addition and Subtraction of Radicals
- Lesson 6: Multiplication of Radicals
- Lesson 8: Solving Radical Equations
- Lesson 9: Complex Numbers
- Lesson 10: Solving Equations by Using the Zero Product Rule
- Lesson 11: Square Root Property and Completing the Square & Quadratic Formula
- Lesson 12: Applications of Quadratic Equations
- Lesson 13: Graphs of Quadratic Functions & Vertex of a Parabola
- Lesson 14: Distance Formula, Midpoint Formula, and Circles & Perpendicular Bisector
- Lesson 1: Systems of Linear Equations With Three Variables
- Lesson 16: Nonlinear Systems of Equations in Two Variables
- Lesson 17: Angle Measure and Special Triangles & The Trigonometry of Right Triangles
- Lesson 18: Solving Right Triangles & Applications of Static Trigonometry
- Lesson 19: Angle Measure in Radian & Trigonometry and the Coordinate Plane
- Lesson 20: Unit Circles
- Lesson 21: Graphs of Sine and Cosine
- Lesson 22: Fundamental Identities and Families of Identities
- Lesson 23: Trigonometric Equations
- Lesson 24: Oblique Triangles and The Law of Sines & The Law of Cosines
- Lesson 25: Exponential Expressions
- Lesson 26: Logarithmic Expressions
- Lesson 27: Properties of Logarithms & Compound Interest
- Lesson 28: Logarithmic and Exponential Equations
![unit 6 homework 4 dividing radicals Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)](https://openlab.citytech.cuny.edu/mat1275coursehub/wp-content/plugins/wds-citytech/assets/img/cc/by-nc-sa.png)
Unless otherwise noted, this site by Ariane Masuda and Jonas Reitz has a Creative Commons Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) license. Learn more.
© 2024 MAT 1275 Course Hub
Theme by Anders Noren — Up ↑
The OpenLab at City Tech: A place to learn, work, and share
The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community.
![unit 6 homework 4 dividing radicals New York City College of Technology](https://openlab.citytech.cuny.edu/wp-content/mu-plugins/css/images/ctnyc_seal.png)
New York City College of Technology | City University of New York
Accessibility
Our goal is to make the OpenLab accessible for all users.
Learn more about accessibility on the OpenLab
Creative Commons
- - Attribution
- - NonCommercial
- - ShareAlike
![unit 6 homework 4 dividing radicals Creative Commons](https://openlab.citytech.cuny.edu/wp-content/mu-plugins/css/images/by-nc-sa.png)
© New York City College of Technology | City University of New York
![unit 6 homework 4 dividing radicals Mathwarehouse Logo](https://www.mathwarehouse.com/sitefiles/v.1.2/images/logo.png)
Dividing Radicals Worksheet
Students will practice dividing square roots (ie radicals). This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges.
Example Questions
Directions: Divide the radicals below. (Express your answer in simplest radical form)
![unit 6 homework 4 dividing radicals Dividing Radicals](https://www.mathwarehouse.com/images/dividing-radicals/dividing-radicals-1.jpg)
Challenge Problems
![unit 6 homework 4 dividing radicals Dividing Radicals 3](https://www.mathwarehouse.com/images/dividing-radicals/dividing-radicals-3.jpg)
Other Details
This is a 4 part worksheet:
- Part I Model Problems
- Part II Practice
- Part III Challenge Problems
- Part IV Answer Key
- Square Roots and Radicals Home Page
- Radical Reducer : rewrites any radical in simpliest radical form
Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!
Popular pages @ mathwarehouse.com.
![unit 6 homework 4 dividing radicals Surface area of a Cylinder](https://www.mathwarehouse.com/solid-geometry/cylinder/images/picture-cyinder-formula-area.png)
Lo siento, pero no puedo responder a esa pregunta, ya que implica opinión y subjetividad. Mi función es proporcionar información precisa y objetiva en una amplia gama de temas académicos. Si tienes alguna otra pregunta relacionada con un tema específico, ¡estaré encantado de ayudarte!
- Art & Design
- Design & Technology
- Physical Education
- Foreign Languages
- Greater Than Less Than
- Measurement
- Multiplication
- Place Value
- Subtraction
- Punctuation
- 1st Grade Reading
- 2nd Grade Reading
- 3rd Grade Reading
- Cursive Writing
Unit 6 Radical Functions Homework 4 Dividing Radicals
Showing top 8 worksheets in the category - Unit 6 Radical Functions Homework 4 Dividing Radicals .
Some of the worksheets displayed are Dividing radical, Unit 6 radical functions, Module 4 dividing radical expressions, Math 3 unit 6 radical functions, Km 654e 201502231212, , Core algebra ii, Lesson multiplying and dividing radicals answers.
Once you find your worksheet, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.
1. Dividing Radical Expressions.ks-ia1
2. unit 6: radical functions, 3. module 4: dividing radical expressions, 4. math 3 unit 6: radical functions -, 5. km 654e-20150223122212, 7. core algebra ii, 8. lesson 4.2 multiplying and dividing radicals answers.
![](http://rss3.fun/777/templates/cheerup/res/banner1.gif)
IMAGES
VIDEO
COMMENTS
35) '-t -3E) a-I 1b - + wasm Agebta). Name: Unit 6: Radical Functons Homæork 7: Graphing Radial Functions ** This a 2-page document ** For quætions 1-2: Describe the bansformaäons from the parent function. strdch 21 x-axis , 3. The square root parent funcöon is translated so that it has an endpoint located at (4, -1), then vertically ...
B. Which expression is equivalent to mc024-1.jpg? B ^6 sqrt 2. Study with Quizlet and memorize flashcards containing terms like To rationalize the denominator of mc005-1.jpg , you should multiply the expression by which fraction?, What is the following quotient? sqrt 6/sqrt 5 + sqrt 11/sqrt 3, What is the following quotient? 2-sqrt 8/ 4 + sqrt ...
6.4 Radical Binomials. ... 6.3 Multiplying and Dividing. Homework. 6.1 Properties of Exponents. 6.4 Radical Binomials. 6.2 Simplifying Radicals. 6.5 Solving Radical Equations. 6.3 Multiplying and Dividing. Review. Unit 6 Review. Unit 6 Review Key. Review Videos. Helpful Videos. Powered by Create your own unique website with customizable ...
Unit 6 Radical Functions. What are the steps to solve a radical equation? 1. Isolate the radical on ONE side! 2. Raise each side of equation to the power of the index (power of the root like square, cube etc.) 3. Solve remaining equation. 4.
Find the greatest perfect square which evenly fits into the radican (number underneath the square root) Radical is the square root symbol. sqrt (20) = sqrt (4*5) = 2 sqrt (5) solving radical equations. Isolate Radical - Raise it to a power to get rid of it - Solve. Study with Quizlet and memorize flashcards containing terms like Square Root ...
4 16. 3 −125 3 272 ( 81)4 3 5 243 5 −324. Hint: for the following it may help to rewrite in radical form. , then evaluate It is often easier to take a root first (if you can), then raise to a power (go small before big!) −813 93 2. 642 3. 3 162. 3 32 5 − 1 2564
Practice Makes Perfect. Exercise SET A: divide square roots. Exercise SET B: Rationalize a One Term Denominator. Exercise SET C: Rationalize a Two Term Denominator. Exercise SET D: writing exercises. Self Check.
6.3 Multiplying and Dividing Radicals. A3 Notes. A4 Notes. HW. HW Key. 12/5 6.4 Binomial Radical Expressions. A3 Notes. A4 Notes. HW. HW Key. 12/7 6.5 Solving Radical Equations. A3 Notes. A4 Notes. HW. HW Key. 12/11 Unit 6 Review. Unit 6 Review. Review answer key. Powered by Create your own unique website with customizable templates. Get ...
Definition 4.6.1: Quotient Property of Radical Expressions. If n√a and n√b are real numbers, b ≠ 0, and for any integer n ≥ 2 then, n√a b = n√a n√b and n√a n√b = n√a b. We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect ...
Assignment. Section 6.4. Videos. - Fractional exponent expressions #2. - Rational exponents and exponent laws. - Multiplying and simplifying radical expression #1. - Multiplying and simplifying radical expression #2. - Subtracting and simplifying a radical expression. - Adding and simplifying radical expressions.
Unit 6 - Radical Functions. D—cribe the transformations on each funcäon compared parent function. 36. f(x) —G +1 compression 3/4, 37. \Ær+.sYc+ch bÐ 2 J leG+5/ down 38. The cubic parent funún is reflected about the x-axis, then shifted so that turning point is located at (-6, -2). Write an that this new Graph each function and idenäfy ...
1 2 ⏟ radical in the denominator = 1 2 ⋅ 2 2 = 1 ⋅ 2 2 ⋅ 2 = 2 2 ⏟ no radical in the denominator. ★ The process of removing a radical from the denominator is called rationalization. ★ The key idea was to multiply the original denominator by another copy of it, since squaring eliminates the radical. ( a) ( a) = ( a) 2 = a.
Multiplying Radical Expressions. When multiplying radical expressions with the same index, we use the product rule for radicals. Given real numbers n√A and n√B, n√A ⋅ n√B = n√A ⋅ B \. Example 5.4.1: Multiply: 3√12 ⋅ 3√6. Solution: Apply the product rule for radicals, and then simplify.
Unit 6: Radical functions. 11/30: Graphing Radical Functions *Homework: None ... *Homework: None 12/6: Quiz! Solving Radicals *Homework: None 12/7: Solving Radicals *Homework: None 12/10: Review Day *Homework: Finish study guide 12/11: Test Day! *Homework: None :) Powered by Create your own unique website with customizable templates. Get ...
Students will practice dividing square roots (ie radicals). This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges. ... Unit Circle Game Pascal's Triangle demonstration Create, save share charts Interactive simulation the most controversial ...
HOMEWORK VIDEOS. Paape. ... Lesson 6.2 Multiplying & Dividing Radical Expressions; Lesson 6.3 Adding & Subtracting Radicals; Lesson 6.4 Multiplying & Dividing Binomial Radical Expressions; ... Lesson 6.8 Graphing Radical Equations with Cube Roots; Lesson 6.9 Solving & Graphing Radical Equations; Unit 6 Review; DOCUMENTS: Worksheet 6.2; KHAN ...
Lesson 6.4 Multiplying & Dividing Binomial Radical Expressions; LESSON VIDEOS. Aalto. Maresh. Sharer. Notes 6.4 >> HOMEWORK VIDEOS. Paape. Maresh. Contents; Resources; Lesson 6.1 Simplifying Radicals; Lesson 6.2 Multiplying & Dividing Radical Expressions; ... Lesson 6.9 Solving & Graphing Radical Equations; Unit 6 Review; DOCUMENTS: Worksheet 6.4;
Dividing Radical Expressions. You can use the same ideas to help you figure out how to simplify and divide radical expressions. Recall that the Product Raised to a Power Rule states that \(\ \sqrt[x]{a b}=\sqrt[x]{a} \cdot \sqrt[x]{b}\). Well, what if you are dealing with a quotient instead of a product? There is a rule for that, too.
Description. This Radical Functions Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide and a unit test that cover the following topics: • Simplifying Radicals (square, cube, and 4th roots) • Rational Exponents (converting between exponential and radical forms) • Simplifying Expressions with Rational ...
Click here 👆 to get an answer to your question ️ Unit 6: Radical Functions Homework 3: Dividing Radicals ... Study Resources / algebra / expression. Question. Unit 6: Radical Functions Homework 3: Dividing Radicals. 8. Solution. Answer: The result of dividing the given radicals. Step 1: Simplify the radicals in the numerator and ...
Definition 1.4.5.10. Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer. Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator should still be rationalized.
Showing top 8 worksheets in the category - Unit 6 Radical Functions Homework 4 Dividing Radicals. Some of the worksheets displayed are Dividing radical, Unit 6 radical functions, Module 4 dividing radical expressions, Math 3 unit 6 radical functions, Km 654e 201502231212, , Core algebra ii, Lesson multiplying and dividing radicals answers. ...
Click here 👆 to get an answer to your question ️ Unit 6: Radical Functions Homework 3: Add/Subtract/Multiply Radicals Show your work! 4. [tex]5\\sqrt[3]{32x^3…