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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Multiply & Divide Radicals

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
Addition
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
Addition Property of Zero
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
II. Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
Contradiction Cases
1:30
Solving Linear Equations in One Variable Cont.
2:00
Addition Property of Equality
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
Addition Property of Equality
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
III. Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
Quadrants
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
IV. Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
V. Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
VI. Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
VII. Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16
Adding & Subtracting Polynomials

18m 33s

Intro
0:00
Objectives
0:07
Adding and Subtracting Polynomials
0:25
Terms
0:33
Coefficients
0:51
Leading Coefficients
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
Adding and Subtracting Polynomials Cont.
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
VIII. Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12
IX. Quadratic Equations
Solving Quadratic Equations by Factoring

23m 38s

Intro
0:00
Objectives
0:08
Solving Quadratic Equations by Factoring
0:19
Quadratic Equations
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22
Solving Quadratic Equations

29m 27s

Intro
0:00
Objectives
0:12
Solving Quadratic Equations
0:29
Linear Factors
0:38
Not All Quadratics Factor Easily
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
The Quadratic Formula
7:28
Discriminants
8:25
Solving Quadratic Equations - Summary
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07
Equations in Quadratic Form

16m 47s

Intro
0:00
Objectives
0:08
Equations in Quadratic Form
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14
Quadratic Formulas & Applications

29m 4s

Intro
0:00
Objectives
0:09
Quadratic Formulas and Applications
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
Quadratic Formulas and Applications Cont.
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50
Graphs of Quadratics

26m 53s

Intro
0:00
Objectives
0:06
Graphs of Quadratics
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
Graphing in Quadratic Standard Form
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
Graphs of Quadratics Cont.
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
X. Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43
Adding & Subtracting Rational Expressions

20m 24s

Intro
0:00
Objectives
0:07
Adding and Subtracting Rational Expressions
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
Subtracting vs. Adding
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
Both Methods Lead to the Same Answer
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
XI. Radical Equations
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
Product Rule for Radicals
0:26
Product Rule to Simplify Square Roots
1:11
Quotient Rule for Radicals
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09
Adding & Subtracting Radicals

17m 22s

Intro
0:00
Objectives
0:07
Adding and Subtracting Radicals
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
Add Radicals' Coefficients
3:55
Example 1
4:47
Example 2
6:00
Adding and Subtracting Radicals Cont.
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10
Multiply & Divide Radicals

19m 24s

Intro
0:00
Objectives
0:08
Multiply and Divide Radicals
0:25
Rules for Working With Radicals
0:26
Using FOIL for Radicals
1:11
Don’t Distribute Powers
2:54
Dividing Radical Expressions
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
Multiply and Divide Radicals Cont.
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
Multiply and Divide Radicals Cont.
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25
Solving Radical Equations

15m 5s

Intro
0:00
Objectives
0:07
Solving Radical Equations
0:17
Radical Equations
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
Solving Radical Equations Cont.
7:04
Solving Radical Equations with More than One Radical
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Lecture Comments (5)

1 answer

Last reply by: Rhonda Steed
Tue Apr 22, 2014 12:20 AM

Post by Deepa Kumar on April 11, 2014

at 15:20 you said 16 minus the square root of 9 equals 16 minus 9 and that equals 7 while the correct answer is 16 minus 3 which equals 13.

2 answers

Last reply by: Mohamed Elnaklawi
Sat Apr 19, 2014 9:56 AM

Post by Mohamed Elnaklawi on April 8, 2014

what does foil stand for?

Multiply & Divide Radicals

  • When multiplying radical expressions of the same index we can combine them into a single radical.
  • Watch for situations that we need to use FOIL in the multiplication process. These situations involve two binomials.
  • To divide by a radical with rationalize its denominator. This means we multiply the top and bottom by an expression that completes the rational in the denominator. There should no longer be any radicals in the denominator.
  • When dividing by a binomial that has a rational expression use the conjugate of the expression. The conjugate is almost the same, but is connected by a different sign.

Multiply & Divide Radicals

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Objectives 0:08
  • Multiply and Divide Radicals 0:25
    • Rules for Working With Radicals
    • Using FOIL for Radicals
    • Don’t Distribute Powers
    • Dividing Radical Expressions
    • Rationalizing Denominators
  • Example 1 7:22
  • Example 2 8:32
  • Multiply and Divide Radicals Cont. 9:23
    • Rationalizing Denominators with Higher Roots
  • Example 3 10:51
  • Example 4 11:53
  • Multiply and Divide Radicals Cont. 13:13
    • Rationalizing Denominators with Conjugates
  • Example 5 15:52
  • Example 6 17:25

Transcription: Multiply & Divide Radicals

Welcome back to www.educator.com.0000

In this lesson let us take a look at how you can multiply and divide radicals.0003

We will first cover some rules for multiplying and dividing radicals and then get into that division process0011

and see how we want to rationalize the denominator that involves a radical expression.0018

There are only a few rules that you have to remember when working with radicals and the good news is we have seen many of these already.0027

For example, you can change radicals into fractional exponents.0033

If you want to combine them for division you can separate those out if you want to separate them out over multiplication.0039

You can also add and subtract radicals as long as we make sure that the radicands and the indexes are exactly the same.0050

Radicals follow our properties for all the other types of numbers so we can also use0059

the commutative property, associative property, even on these radicals.0064

The reason why that is important is because there is a few situations where you often try to apply some rules that do not work.0072

Treat these radicals like they are any other numbers.0080

Let us see this as we walked through the following problem.0084

I want to multiply the √6 + 2 × √6-30087

For this one treat it like binomials and use foil to multiply everything out.0093

Every terms multiplied by every other term.0099

Let us see our first terms would have √6 × √6, outside terms -3 √6, inside terms 2√6 and our last terms 2 × -3 = 6.0103

Then we can use our other rules to go ahead and simplify this further.0119

I see I have two roots here, I can put those underneath the same root and that is the same as the √36 or 6.0125

-3 × √6 + 2 × √6 can I add them together or not?0135

Yes I can because they have the same radical I will just do the coefficients.0142

-3 + 2 - √60147

- 6 is still there.0157

You can continue simplifying your like terms by combining the 6 and -6 giving -√6.0159

We are using many of our tools that we have learned up to this point, in order to handle these radicals.0168

Watch out for lots of situations where you need to use foil.0176

In this example I have 4 + √7 2 it is tempting to try and take that 2 and distribute it over the parts in between.0181

However, do not do that.0191

Do not even attempt any type of distribution with this one.0194

What you should do with it is foil because as we learned in our exponent section, this stands for 4 + √7 and 4 + √7 .0197

Those two things are being multiplied by each other.0208

I can see where foil comes in to play.0212

I will take my first terms 16, outside terms, inside terms, and last terms.0216

Unlike before I could use some other rules to go ahead and combine things.0227

I can go ahead and add these two since they have the same radical and get 8√7.0232

I can combine these under the same radical and get the √49 and that is simply 7.0239

We can go ahead and finish off this problem, 16 + 7 would be 23 or 23 + 8 × √7.0249

Let us take a close look at division.0266

When dividing a radical expression we go ahead and rewrite it by rationalizing the denominator.0269

If you have never heard that term before rationalizing the denominator,0278

it is a way of rewriting it so that there is no longer a radical in the bottom.0281

That seems a little weird.0286

I mean, if we are interested in dividing by radical why are we writing it that there is no radical in the bottom.0287

It seems like we are side stepping the problem like we are not ending up dividing by the radical.0294

It is just a simpler way of looking at the whole division process and it is something that you have done before with fractions.0299

For example, when we have 1/3 ÷ 2/5 you have been taught that you should always flip the second and then multiply.0307

Why are we doing that? Why do not we just go ahead and divide by 2/5?0314

What is the big deal with a turn it into multiplication problem?0317

One, we will show you how accomplish the same thing, but by flipping the second one multiplying it does it in a much simpler way.0321

Also supposed I write this problem as 1/3 ÷ 2/5, you recognize that this is one of our complex fractions.0330

I can simplify a complex fraction by multiplying the top and bottom by the same thing.0341

I will multiply the top and bottom by 5/2, that should be able to get rid of our common denominator.0348

On top I would have 1/3 × 5/2 and on the bottoms 2/5 × 5/2 would be 1.0359

What we are sitting on the top there is the 1/3 and there is the 5/2 which comes from that rule we learned.0369

That we should take the second one, flip it and multiply.0379

But notice how we are doing that by simplifying the bottom now we are dividing by 1.0382

It is a great way to end up rewriting the problem, and taking care of it in a much simpler way.0390

That is exactly what we want to do when we are rationalizing.0396

The actual steps for rationalizing the denominator look a lot like this.0402

First, we will end up rewriting the rational expression, so that we will end up with no root in the bottom.0407

When we rationalize we try to get rid of that root.0413

We will do this by multiplying the top and bottom by the smallest number that gets rid of that radical expression.0416

That sometimes you can use some larger numbers, but it is best to use the smallest thing that will get rid of that radical.0422

It saves you from doing some extra simplifying in the end.0429

If we are dealing with square roots I recommend try and make that perfect square0434

in the denominator and that should be able to rationalize it just fine.0437

Let us see one of this division by radical in process and this is also known as rationalizing the denominator.0444

I have 2 ÷ √(2 )0451

I’m going to end up rewriting this so that there is no longer √2 in the bottom.0456

We will do this by multiplying the top and bottom by another root so that we will have a squared number in the root for the bottom.0462

On top I have 2√(2 ) and in the bottom I can go ahead and put these together and get √2 × 2, which is the √4.0471

I have created that square number on the bottom and now we can go ahead and simplify it.0485

That will be 2.0493

If you can simplify from there go ahead and do so, you will see that this one turns into the √2.0496

When I take 2 ÷√2 like the original problem says the √2 is my answer.0503

Let us try another one.0513

This one is 12 ÷ √50515

We are looking to multiply the top and bottom by something to get through that √5 in the bottom we will use another√5.0519

12 √5 for the top and bottom √5 × √5 would be the same as √25.0529

The good news is that one reduces and becomes just 5.0541

If you are doing a division, you are dividing by the √5 even though it looks like you are changing into multiplication problem.0549

This is also known as rationalizing the denominator.0558

You can use these tools like rationalizing the denominator for some much higher roots as well.0565

The thing to remember by is that you want to multiply by the smallest root actually complete set root.0571

When we are dealing with the square roots it look like always multiplying by the same thing on the bottom.0577

With high roots, sometimes that might not be the case.0583

Let us look at this one.0586

1 / 3rd root of 2 if I try and multiply the top and bottom by 3rd root of 2 it is not going to get rid of that radical.0587

We will be left with the 3rd root of 2 / 3rd root of 4 and 4 is not a cubic number.0598

It is like it did not have enough of the number on the bottom to go ahead and simplify it completely.0604

Which should we multiply?0612

If I want a cubic number in the bottom but I'm going to use 3rd root of 4 .0615

When I take that onto the top and bottom you will see that indeed we do get that cubic number that we need.0625

From the bottom this would be 3rd root of 8 and on the top I will just have 3rd root of 4 .0632

The bottom simplifies becomes 2 now my answer is 3rd root of 4 ÷ 2.0640

Let us try another one of those higher roots.0652

This one is 3 ÷ 4th root of 90654

Let us see, what would I have to use with a 4th root of 9?0659

9 is the same as 3 and 3, it will be nice if I had even more 3’s underneath there.0664

Let us say a couple of more.0671

I will accomplish that by doing the top and bottom by 4th root of 9 .0674

3 × 4th root of 9 for the top and 4th root of 81 on the bottom.0682

Now it is looking much better.0693

34th root of 9 on the top, the bottom simplifies to just the 3 and now we will cancel out these extra 3s.0696

I have 4 4th root of 9 .0706

This one we want to rationalize the denominator and if you look at it you must think what denominator are we trying to rationalize?0715

Is there a root in the bottom?0722

Because of our rules that allow us to break up the root over the top and bottom, there is.0725

In this one we have the 3rd root of 2y / 3rd root of z.0731

We can see we have 3rd root of z and it definitely needs to be simplified.0744

How are we going to do that?0750

Since I'm dealing with a cubed root I will need an additional 2 z's for that bottom.0752

Let us use the cube root of z2.0758

Watch what that would give us here on the bottom.0765

3rd root of z3 on top, the 3rd root of 2y z2.0768

The bottom simplifies and there will no longer be any roots in the bottom.0779

3rd root of 2y z2 / z and I know that this one is done.0784

One thing that can make the rationalization a more difficult process since we have more than one term in the bottom.0797

Our main goal is to end up rewriting the bottom so that there is no longer a root.0805

If we have more than one term we are going to end up using something known as0812

the conjugate of the expression to go ahead and get rid of it.0815

What the conjugate is, it is the same as our original expression, but it has a different sign connecting them.0820

That will allow us to get rid of that root.0827

To show you why we get through the root we will use an example.0830

Here I have 4 + √3, the conjugate of this one would be the same I have a 4√3 it will have a different sign connecting them 4 - √3.0835

Watch what happens when I foil these two together.0852

The 4 + √3 and its conjugate.0855

Starting with the first terms I have 4 × 4, which would be 16.0863

My outside terms would be -4√3, my inside terms will be 4√3.0868

I can move on to my last terms -√3 × √3.0877

A lot of things are happening when you multiply by its conjugate.0882

One, notice how our outside and inside terms where the same but one was positive and one was negative.0886

When you are dealing with conjugates that should always happen.0892

Those two things are gone.0897

We will focus was going on down here on the end.0899

-√3 × √3, -3 × 3 which be 16 - √9 which would be 9.0903

The numbers may change to make it different but at this point, there is no more radical.0918

Because those two radicals multiply and I get that perfect square number, there is no more radicals to deal with.0925

I can just take 16 - 9 and get a result of 7.0932

By multiplying by that conjugate, I got rid of all instances of all radicals.0940

This is why it will be important to use it when getting rid of our radical on the bottom.0946

Let us see for these examples.0953

Notice we need that conjugate because we have two terms in the denominator.0955

I'm going to multiply the top and bottom of this one by the conjugate.0962

√5 – 2 and √5 - 20968

When dealing with more than one term, remember that you will have to foil out the bottom.0976

Also remember that though you will have to possibly foil or even distribute the top.0984

The top 9√5 – 18 and working with the bottom and foiling that out my first terms would be √5 × √5 =5.0991

Outside terms and inside terms they are going to cancel out I know I’m on the right track.1006

2 × -2 -41015

We will go ahead and do some canceling and let us see what we have left over.1020

9√5 - 18 / 5 – 4 = 1 and this reduces to 9√5 - 18.1024

Notice how we have divided by that radical because we have gotten rid of that radical in the bottom.1037

In this last example, let us look at rationalizing the following denominator.1046

We have 7 ÷ 3 - √x1050

With this one I’m going to have to multiply the top and the bottom by its conjugate.1055

You know what is next in there, this will be 3 + √x we will do that on the bottom and on the top.1064

Let us see what this does to the top as we distribute and remember that on the bottom we will foil.1073

I get 21 + 7√x for the top.1082

On the bottom we have 9 + 3 × √x - 3 × √x and then - √x × √x.1089

If we do things correctly we should get rid of all those radicals in the bottom.1108

+ square root - square root those two will take care of each other and then my √x× √x = x.1114

This will leave us with 21 + 7 √x / 9 – x.1123

We have got rid of all those radicals in the bottom you can say that our denominator is rationalized.1133

You can see that when you are working with dividing radicals you always have to keep in mind1139

what you would put on the bottom in order to get rid of all those radicals.1144

If you only have a single term feel free to multiply by what would complete whatever that radical is.1148

Complete the square or complete the cube.1154

If you have more than one term use the conjugate to go ahead and rationalize the denominator.1157

Thank you for watching www.educator.com.1162

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