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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Adding & Subtracting 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 (1)

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Post by Keisha Jordan on October 27, 2015

vvvv vvvv

Adding & Subtracting Radicals

  • Radical expressions can only be added and subtracted if they have the same radicand, and index.
  • Some radicals must be re-written before you can add or subtract them. This is done by simplifying the roots first.
  • If it is possible to add or subtract to terms with radicals, then we add and subtract their coefficients. This is similar to adding and subtracting like terms.

Adding & Subtracting Radicals

Simplify:
3√7 + 8√5 − 9√7 − 6√5
  • ( 3√7 − 9√7 ) + ( 8√5 − 6√5 )
− 6√7 + 2√5
Simplify:
4√{11} − 20√3 − 16√{11} − 5√3
  • ( 4√{11} − 16√{11} ) + ( 20√3 − 5√3 )
− 12√{11} − 25√3
Simplify:
64√{17} − 12√2 − 23√2 + 15√{17}
  • ( 64√{17} − 15√{17} ) + ( 12√2 − 23√2 )
79√{17} − 35√2
Simplify:
3√{12x} + 4√{27x} − 5√{3x} + 6√{75x}
  • 3√4 ×√{3x} + 4√9 ×√{3x} − 5√1 ×√{3x} + 6√{25} ×√{3x}
  • 3 ×2√{3x} + 4 ×3√{3x} − 5 ×1√{3x} + 6 ×5√{3x}
  • 6√{3x} + 12√{3x} − 5√{3x} + 30√{3x}
43√{3x}
Simplify:
7√{32y} − 8√{72y} + 4√{98y} + 6√{200y}
  • 7√{16} ×√{2y} − 8√{36} ×√{2y} + 4√{49} ×√{2y} + 6√{100} ×√{2y}
  • 7 ×4√{2y} − 8 ×6√{2y} + 4 ×7√{2y} + 6 ×10√{2y}
  • 28√{2y} − 48√{2y} + 28√{2y} + 60√{2y}
68√{2y}
Simplify:
5√{196n} − 3√{256n} − 2√{324n} + 8√{36n}
  • 5√{49} ×√{4n} − 3√{64} ×√{4n} − 2√{81} ×√{4n} + 8√9 ×√{4n}
  • 5 ×7√{4n} − 3 ×8√{4n} − 2 ×9√{4n} + 8 ×3√{4n}
17√{4n}
Simplify:
( 4√3 − 6√5 )2
  • ( a − 6 )2 = a2 − 2ab + b2
  • ( 4√3 )2 − 2( 4√3 )( 6√5 ) + ( 6√5 )2
  • 16 ×3 − ( 2 )( 4 )( 6 )√3 √5 + 36 ×5
  • 48 − 48√{3 ×5} + 180
228 − 48√{15}
Simplify:
( 2√7 − 3√{11} )2
  • ( 2√7 )2 − 2( 2√7 )( 3√{11} ) + ( 3√{11} )2
  • 4 ×7 − 2( 2 )( 3 )√7 √{11} + 9 ×11
  • 28 − 12√{7 ×11} + 99
127 − 12√{77}
Simplify:
( 2√{10} − 4√{12} )( 3√{15} − 5√5 )
  • ( 2√{10} )( 3√{15} ) + ( 2√{10} )( − 5√5 ) + ( − 4√{12} )( 3√{15} ) + ( − 4√{12} )( − 5√5 )
  • 6√{10} √{15} − 10√{10} √5 − 12√{12} √{15} + 20√{12} √5
  • 6√{150} − 10√{50} − 12√{180} + 20√{60}
  • 6√{25 ×6} − 10√{25 ×2} − 12√{9 ×4 ×5} + 20√{4 ×15}
  • 6 ×5√6 − 10 ×5√2 − 12 ×3 ×2√5 + 20 ×2√{15}
30√6 − 50√2 − 72√5 + 40√{15}
Simplify:
( 6√{40} + 7√{24} )( 10√{18} − 12√{30} )
  • ( 6√{40} )( 10√{18} ) + ( 6√{40} )( − 12√{30} ) + ( 7√{24} )( 10√{18} + ) + ( 7√{24} )( − 12√{30} )
  • 60√{720} − 72√{1200} + 70√{432} − 84√{720}
  • 60√{9 ×16 ×5} − 72√{100 ×4 ×3} + 70√{16 ×9 ×3} − 84√{9 ×16 ×5}
  • 60 ×3 ×4√5 − 72 ×10 ×2√3 + 70 ×4 ×3√3 − 84 ×3 ×4√5
  • 720√5 − 1440√3 + 840√3 − 1008√5
− 288√5 − 600√3

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

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

Transcription: Adding & Subtracting Radicals

Welcome back to educator.com.0000

In this lesson we are going to take a look at adding and subtracting radicals.0003

You may notice in a lot of my other lessons I avoid trying to add or subtract radicals as much as possible.0009

It is because there is a few problems that you will run into when you try and these radical expressions.0015

Look at some of the problems that we want to avoid.0020

Do not get into the actual rule for adding and subtracting that way you will know0023

what situations that you can add and subtract and which situations you cannot.0027

A lot of the other rules that we have picked up for radicals so far we have been doing lots of multiplication and division.0035

You will notice that in those rules they follow pretty much exactly from the rules of exponents.0042

There is a nice product rule, quotient rule, and they seem to mimic one another.0048

To understand some of the difficult things that we run into with adding, subtracting we have to look at what happens0053

when some of our rules for adding, subtracting when you have things with exponents.0060

For example, I want to put together x2 + 3x what problems when I run into? can I put those two things together or not?0064

You will notice that you will run into quite a bit of a problem.0075

These ones do not have the same exponents, I cannot put them together.0079

Since the terms are not like terms and they have different exponents maybe we can change it around and try a different situation.0084

Let us go ahead and try ab2 + r2.0100

Can we put those together? After all the exponents here are exactly the same.0104

Can we combine those like terms?0110

These one are not like terms.0114

We ran into a very similar problem the bases are not the same with these two.0115

Those ones are not the same, they do not have the same base then we cannot put them together.0124

We usually run into one of those two problems where we are dealing with radicals.0131

Either they do not have the same base or they do not have the same exponents.0135

Here is a quick example involving radicals so we can see what I’m talking about.0139

Here I have 3rd root of x + √30142

If I write those as exponents, then it is like x1/3 + 3^½.0147

These have different bases and they have different exponents, there is no reason why you should be able to put these together.0153

After seeing many of these different examples, you might that be under the false assumption that we cannot add any radicals together.0163

In some instances you will be able to put these radicals together you have to be very careful on certain conditions.0173

One, when putting radicals together you have to make sure that their powers, those would be the indexes of each0182

of the radicals are the same and you have to make sure that the radicand or the bases are exactly the same.0187

I can put together the 3rd root of 5x with the other 3rd root of x.0194

It is completely valid because when written as exponents I have the bases the same and they are both raised to the power of 1/3.0200

How would I go about actually putting them together?0210

I will treat them just almost like an entire variable.0213

If I was adding u + u I will get two u.0216

Notice how this common pieces here.0221

Another way of saying that is, we simply add together our coefficients out front so one of those should equal two of them.0224

Make sure that you keep in mind that if you are going to add and subtract radicals0237

you must have the same index on those radicals and you must have the same radicand.0242

That is the part underneath radicals.0246

Once you get to the addition or subtraction process, look at your coefficients out front, so (5 × √x) + (3 × √x) – (6 × √x) .0248

I’m looking at 5 + 3 - 6.0260

That will give me a result of 2.0265

This is exactly the same process that you would go through if you are just adding like terms.0270

That will be u2 + 7u2 - u2 then you are just looking at these initial coefficients like 2 + 7 - 1, and that would give you the 8.0275

Let us see if we can take a look at some examples on when we can add and subtract these radicals.0288

In the first one I'm looking at 3 × 4th root of 17 – 4th root of 170294

Let us check, the indexes are exactly the same we are looking at the 4th root and our radicands.0300

That is the part underneath, they are both 17.0308

We are going to look at the coefficients, 3 - 1 = 2.0312

I have 2 4th root of 17 and that one is good.0318

Let us look at this other one.0326

21√a + 4 3rd root of a0328

It is tempting to want to put these ones together but we cannot do it.0335

This one is the square root and this one is a cubed root.0340

It must have the same index if you have any hope of getting those together.0353

Let us look at some others.0357

On this one we want to add or subtract if possible.0361

I have 3 + √xy + 2 × √xy0365

Both of these are dealing with square roots and that is good.0373

Both of these are with an xy underneath that root.0376

We will simply add together their coefficients.0381

This will give us a 5√xy.0386

Let us see how that works for the next one.0392

7 × 5th root of u3 - 3 × 5th root of y3.0393

That is so close.0400

Both of them have a 5th root and things are being raised to the third power, all of that is matching up but the variables are completely different.0402

One is a u and one is a y.0412

Make sure you check your indexes and you check your radicand before you ever put them together using addition or subtraction.0420

There are a few instances where the indexes are the same, but it looks like that radicand on the part underneath is completely different.0431

It is tempting to write those off and say okay, I probably cannot put those together using addition or subtraction.0440

Sometimes if you can do a little bit of simplification and get them the same then you can go ahead and put those together.0446

Let me show you an example of numbers.0453

Suppose I wanted to put together √2 + √8 and just looking at them I will say that wait 8 and 2 they are not the same, I cannot put them together.0455

The √8 over here that is the same as 4 × 2 and I can take √4 and that would leave me with 2 × √2 .0466

I can simply rewrite the next one as 2 × √2 .0480

In doing so now my radicals are exactly the same and I can simply focus on these coefficients out front.0485

I can see that 1 + 2 does equal my 3.0493

Do not be afraid to try and simplify these a little bit before you get into the addition or subtraction process.0497

Let us try that and keep it in line with these ones.0504

We want to rewrite the expressions and then try and add or subtract them if possible.0507

The first one I have a -√ 5 + 2 × √125.0513

I have -√ 5, 2 and 125 if I want to end up rewriting that, that is a 5 × 25 so that one reduces.0520

I have the √5 × 5.0542

Let us write that as √ 5 + 10√ 5 and now that I have my radicals the same now just focus on this coefficients -1 + 10 would be 9√ 5.0548

The next one I chose a big number but no worries, we can take care of this one.0568

We are looking for the 4th root of 3888 + 7 × 4th root of 30575

If I have any hope of putting these together I want to match this 4th root of 3 over here.0584

As I go searching for ways to break down that very large number, the very first thing I'm going to try and break it down with is 3.0591

Let us see if I can.0600

It is the same as 1296 × 3 that is good because if I look at 1296 I can take the 4th root of that and I will get 6.0608

I simply have to add together these other radicals here by looking at their coefficients.0631

6 + 7 = 13 4th root of 30637

Let us try one more, √72x - √32x0648

Let us try and simplify this as much as possible.0655

With the first one, looking at 72 is the same as 36 × 2 and with 32 that 16 × 2.0659

Notice how I have the square numbers underneath here, but I can go ahead and simplify.0674

√36 that will be 6 and I still have that 2 underneath there, √16 will be 4 and there is the 2x for that one.0679

I’m looking at 6√2x -4 × √2x or 2 √2x0691

These ones are a little bit larger involving some much higher roots, but the same process applies.0708

We must get the part underneath the roots the same if we are going to be able to put these two together.0714

10 × 4th root of m3 is already broken down as far as it will go.0724

The next one I could look at the 6561 and try and take its 4th root and break it apart from its m3.0736

The good news is that one does break down, you will get 9.0751

4th root of m30763

10 4th root of m3 + 9d4th root of m3 and we could put those together 10 and 90 is 1004th root of m3 .0767

Let us try this next one here, this one is the 3rd root of 63xy2 – 3rd root of 125x4y50789

In this one we will not only need to simplify those numbers but also take care of the variables like the x and y.0799

First, the numbers the 3rd root of 64 what does that break down into?0811

That will go in there 4 times and I have a 3rd root of xy2,0820

that one does not break down any further because both of those powers are smaller than the index of 3.0827

The 3rd root of 125 would be 5 and let us see what we can do with those variables.0834

X3 × x that will be x4 and y3 × y2 that would give me y5.0843

This right here I can go ahead and take out of radicals.0855

Okay, taking out the x, taking out the y, and I still have xy2.0865

Things are looking good and the part underneath the radical is now exactly the same, we will worry about our coefficients.0873

Notice how our coefficients are not like terms, I will be able to write them simply as 4 - 5xy package them together and then write my radical.0880

It is like we are just factoring out this common piece and writing it outside here.0895

In all cases, make sure you get those radicals exactly the same and combine their coefficients.0904

One last example that we can see many of our different rules in action, we will try and combine 3rd root of 2 / x12 - 3 × 3rd root of 3 / x15.0912

Starting off, I'm going to use my quotient rule to break that up over the top and over the bottom.0927

We will break this one up over the top and over the bottom as well, looks pretty good.0938

I will go ahead and simplify the roots on the bottom.0948

23rd root of 2 / x4 - 3 × 3rd root of 3 / x5.0953

Since we are looking to combine things as much as possible,0967

I will get a common denominator by putting an x on the bottom and on the top for the left fraction.0969

2x 3rd root of 2/x5 - 3 3rd root of 3 /x5.0979

As I continue trying to put them together here is one of those situations where we are stuck, we cannot move any farther from there.0993

Since this is 3rd root of 2 and this is the 3rd root of 3 and those are different.1000

You will know it is tempting but we cannot put them together anymore.1006

We will leave this as 2x3rd root of 2 - 33rd root of 3/x5.1011

A lot of different rules to keep track of our radicals but as long as you remember the rules follow directly from the rules for exponents you should be okay.1021

Be very careful in adding and subtracting those radicals and make sure everything is satisfied before you even attempt to put them together.1031

Thank you for watching educator.com1040

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