Enter your Sign on user name and password.

Forgot password?
Sign In | Subscribe
INSTRUCTORS Carleen Eaton Grant Fraser
Start learning today, and be successful in your academic & professional career. Start Today!

Use Chrome browser to play professor video
Dr. Carleen Eaton

Dr. Carleen Eaton

Operations with Radical Expressions

Slide Duration:

Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
Loading...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Algebra 2
  • Discussion

  • Study Guides

  • Practice Questions

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books

Lecture Comments (5)

0 answers

Post by will samfield on February 9, 2016

Nvm i fixed it sorry

0 answers

Post by will samfield on February 9, 2016

Your videos arnt working

0 answers

Post by julius mogyorossy on November 18, 2014

Dr. Carleen, I discovered a rational number for Pi. Thanks for the educating.

1 answer

Last reply by: Dr Carleen Eaton
Sun Mar 11, 2012 6:53 PM

Post by Jeff Mitchell on March 4, 2012

In the first review example, it looks like toward the end of the example you left off the ^3 for x. so it should have been xyz*[fifth-root](x^3y^4) / 3

Operations with Radical Expressions

  • To simplify a square root expression, extract all perfect squares from the radicand.
  • When adding or subtracting, combine only like radicals.
  • To rationalize a radical expression with a binomial denominator, multiply the numerator and denominator by the conjugate of the denominator.

Operations with Radical Expressions

Simplify: [(√2 − √3 )/(2 + √3 )]
  • 1) Multiply the numerator and denominator by the conjugate of the denominator.
  • Use the shortcut a2 − b2 given a and b
  • [(√2 − √3 )/(2 + √3 )]*[(2 − √3 )/(2 − √3 )]
  • a = 2;b = √3
  • 2) Multiply in the numerator
  • = [(( √2 − √3 )(2 − √3 ))/(( 2 )2 − ( √3 )2)] = [(( √2 − √3 )(2 − √3 ))/1] = [(2*√2 − √2 *√3 − 2√3 + √3 √3 )/1]
  • Simplify as much as possible
= [(2√2 − √6 − 2√3 + 3)/1] = 2√2 − √6 − 2√3 + 3
Simplify: [(√5 + √2 )/(2 − √3 )]
  • 1) Multiply the numerator and denominator by the conjugate of the denominator.
  • Use the shortcut a2 − b2 given a and b
  • [(√5 + √2 )/(2 − √3 )]*[(2 + √3 )/(2 + √3 )]
  • a = 2;b = √3
  • 2) Multiply in the numerator
  • = [(( √5 + √2 )(2 + √3 ))/(( 2 )2 − ( √3 )2)] = [(( √5 + √2 )(2 + √3 ))/1] = [(2√5 + √5 *√3 + 2√2 + √2 √3 )/1]
  • Simplify as much as possible
[(2√5 + √5 *√3 + 2√2 + √2 √3 )/1] = 2√5 + √{15} + 2√2 + √6
Simplify: [(√5 − √3 )/(√5 + √3 )]
  • 1) Multiply the numerator and denominator by the conjugate of the denominator.
  • Use the shortcut a2 − b2 given a and b
  • [(√5 − √3 )/(√5 + √3 )]*[(√5 − √3 )/(√5 − √3 )]
  • a = √5 ;b = √3
  • 2) Multiply in the numerator
  • = [(( √5 − √3 )(√5 − √3 ))/(( √5 )2 − ( √3 )2)] = [(( √5 − √3 )(√5 − √3 ))/2] = [(√5 √5 − √3 √5 − √3 √5 + √3 √3 )/2]
  • Simplify as much as possible
  • [(√5 √5 − √3 √5 − √3 √5 + 2√2 + √3 √3 )/2] = [(5 − 2√{15} + 2√2 + 3)/2] = [(8 − 2√{15} )/2] = [(2(4 − √{15} ))/2] = [((4 − √{15} ))/]
4 − √{15}
Simplify: [(√2 − √3 )/(√5 + √2 )]
  • 1) Multiply the numerator and denominator by the conjugate of the denominator.
  • Use the shortcut a2 − b2 given a and b
  • [(√2 − √3 )/(√5 + √2 )]*[(√5 − √2 )/(√5 − √2 )]
  • a = √5 ;b = √3
  • 2) Multiply in the numerator
  • = [((√2 − √3 )(√5 − √2 ))/(( √5 )2 − ( √2 )2)] = [(( √2 − √3 )(√5 − √2 ))/3] = [(√2 √5 − √2 √2 − √3 √5 + √3 √2 )/3]
  • Simplify as much as possible
[(√2 √5 − √2 √2 − √3 √5 + √3 √2 )/3] = [(√{10} − 2 − √{15} + √6 )/3]
Simplify: [(√5 − √2 )/(5 + √3 )]
  • 1) Multiply the numerator and denominator by the conjugate of the denominator.
  • Use the shortcut a2 − b2 given a and b
  • [(√5 − √2 )/(5 + √3 )]*[(5 − √3 )/(5 − √3 )]
  • a = √5 ;b = √3
  • 2) Multiply in the numerator
  • = [((√5 − √2 )(5 − √3 ))/(( 5 )2 − ( √3 )2)] = [((√5 − √2 )(5 − √3 ))/22] = [(5√5 − √5 √3 − 5√2 + √2 √3 )/22]
  • Simplify as much as possible
[(5√5 − √5 √3 − 5√2 + √2 √3 )/22] = [(5√5 − √{15} − 5√2 + √6 )/22]
Simplify: [(√5 − √2 )/(5 + √3 )]
  • 1) Multiply the numerator and denominator by the conjugate of the denominator.
  • Use the shortcut a2 − b2 given a and b
  • [(√5 − √2 )/(5 + √3 )]*[(5 − √3 )/(5 − √3 )]
  • a = √5 ;b = √3
  • 2) Multiply in the numerator
  • = [((√5 − √2 )(5 − √3 ))/(( 5 )2 − ( √3 )2)] = [((√5 − √2 )(5 − √3 ))/22] = [(5√5 − √5 √3 − 5√2 + √2 √3 )/22]
  • Simplify as much as possible
[(5√5 − √5 √3 − 5√2 + √2 √3 )/22] = [(5√5 − √{15} − 5√2 + √6 )/22]
Simplify (2 + √3 )( − 5 + √3 )
  • Multiply using the foil method
  • (2 + √3 )( − 5 + √3 ) = (2)( − 5) + 2√3 − 5√3 + √3 √3
  • Simplify as much as possible
  • = − 10 + 2√3 − 5√3 + 3
= − 7 − 3√3
Simplify (√5 + √3 )(2√5 + √3 )
  • Multiply using the foil method
  • (√5 + √3 )(2√5 + √3 ) = (√5 )(2√5 ) + √5 √3 + (√3 )(2√5 ) + √3 √3
  • Simplify as much as possible
  • = 10 + √{15} + 2√{15} + 3
= 13 + 3√{15}
Simplify 3√2 − 2√{54} + 3√{18}
  • Notice that only the second and third radical can be further simplified.
  • Simplify by finding the prime factors of 54 and 18
  •   − 2√{54}3√{18}
    Reduced Roots − 2*3√63*3√2
      − 6√69√2
  • Simplify a smuch as possible by combining like terms
3√2 − 6√6 + 9√2 = 12√2 − 6√6
Simplify − 3√{45} + 3√5 − 2√{20}
  • Notice that only the first and third radical can be further simplified.
  • Simplify by finding the prime factors of 45 and 20
  •   − 3√{45} − 2√{20}
    Reduced Roots − 3*3√5 − 2*2√5
      − 9√5 − 4√5
  • Simplify a smuch as possible by combining like terms
− 3√{45} + 3√5 − 2√{20} = − 9√5 + 3√5 − 4√5 = − 10√5

*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

Operations with Radical Expressions

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
  • Properties of Radicals 0:16
    • Quotient Property
    • Example: Quotient
    • Example: Product Property
  • Simplifying Radical Expressions 3:24
    • Radicand No nth Powers
    • Radicand No Fractions
    • No Radicals in Denominator
  • Rationalizing Denominators 8:27
    • Example: Radicand nth Power
  • Conjugate Radical Expressions 11:47
    • Conjugates
    • Example: Conjugate Radical Expression
  • Adding and Subtracting Radicals 16:12
    • Same Index, Same Radicand
    • Example: Like Radicals
  • Multiplying Radicals 19:04
    • Distributive Property
    • Example: Multiplying Radicals
  • Example 1: Simplify Radical 24:11
  • Example 2: Simplify Radicals 28:43
  • Example 3: Simplify Radicals 32:00
  • Example 4: Simplify Radical 36:34

Transcription: Operations with Radical Expressions

Welcome to Educator.com.0000

Today, we will be covering operations with radical expressions.0002

And in earlier lessons in Algebra I, we talked about square roots and properties when working with square roots.0006

And now, we are going to go on to talk about other roots.0012

I will be starting out with reviewing properties of radicals and applying these properties0017

to roots other than square roots: cube roots, fourth roots, sixth roots, and on.0021

It is the same properties, such as the quotient property.0025

And the quotient property says that, if you have the root (it could be a square root;0029

it could be a cube root) of a/b, this is equivalent to the root of a divided by the root of b,0034

with the restriction that b cannot be equal to 0, because if b did equal 0,0045

the square root of that would be 0 (or cube root, or whichever root).0050

And then, you would end up with 0 in the denominator, which would result in an undefined expression.0053

OK, so first looking at this quotient property with an example: the fifth root of 17x10, all of that divided by 12y4.0060

This is the fifth root of 17x10, divided by 12y4.0081

This can be split up into this: the fifth root of 17x10 over the fifth root of 12y4.0087

And the quotient property, along with the product property, allows us to simplify radical expressions,0099

which we are going to talk about in a few minutes.0105

Here, I have the product property: this would be something along the lines of0108

the fourth root of 4x6 equals the fourth root of 4, times the fourth root of x6.0114

And again, this helps us to simplify--the same idea as working with square roots, only now the index is a different number.0131

There is also the restriction that, if n is even, a and b are greater than or equal to 0.0139

And that would be to avoid this situation: look at something like the square root of 10.0143

Using the product property, I could say, "OK, this equals the square root of 5 times 2, which equals the square root of 5, times the square root of 2."0150

So, I have an even index, and I end up with this--not a problem.0160

However, I could also say, "Well, the square root of 10...10 also factors out to -5 times -2."0164

So, I follow the product property, and I end up with the square root of -5, times the square root of -2.0175

And this is not what we are looking for when we talk about a radical; we are staying with real numbers.0183

And so, when we have an even index (it could be 2, 4, 6, 8, something higher), then we define a and b as greater than or equal to 0, not as negative numbers.0188

Again, reviewing some concepts that were learned in Algebra I and applied to square roots,0205

but applying them to other roots this time: simplifying: a radical expression is simplified if the index is as small as possible,0210

the radicand contains no nth powers and no fractions, and no radicals are in the denominator.0221

Let's go ahead and look at this part: the radicand contains no nth powers.0227

Starting out with square roots, just to illustrate this: if you have something like the square root of 18,0240

this can be rewritten, using the product property, as 9 (which is 32) times 2.0248

Well, since the root here, the index, is 2, here I have n = 2.0257

I look in the radicand, and I have an nth power; I have 2.0268

So, this could be further simplified to say "the square root of 32, times the square root of 2, equals 3√2."0273

We did this earlier on; but we just restricted this discussion to square roots.0282

I could look at a more complicated example, using an index of 3, looking for the cube root.0287

If I had something like the cube root of 27, times x7: this is not in simplest form,0294

because I have an index of 3, and I could rewrite this as 33...well, x7 is equal to x6 times x.0303

And x6 is equal to (x2)3; so if I rewrite this as x6(x),0318

I could then go on and write this as the cube root of 33,0332

times...since x6 is equal to...(x2)3, times x.0341

OK, so now, I can see that this was not in simplest form, because I have elements here in the radicand that were raised to the n power.0349

So now, what I can do is say, "OK, this 3 essentially cancels out that 3; and I end up with just a 3."0362

The cube root of 33, of 27, is 3; the cube root of x2, cubed, is x2.0369

And that just leaves an x behind, like this.0376

OK, so simplest form means that the radicand contains no nth powers.0380

Look at the index and make sure that you can't factor out something that would be raised to that same power.0385

Also, the radicand cannot contain fractions--no fractions if it is in simplest form.0393

Something such as the fourth root of x, over 2z, is not in simplest form,0404

because there is a fraction under this fourth root sign.0414

And we can use the quotient property to simplify this; and we will talk in a few minutes0422

about how you go about getting rid of fractions that are under the radical sign.0429

OK, the other thing is: no radicals in the denominator.0436

So, if I have something such as 3y divided by the cube root of 2y, this is also not in simplest form.0447

And again, we are going to talk about how to get rid of radicals in the denominator, how to go about simplifying those, in just a second.0455

So, the index is as small as possible; that is what we are going to be working with (this is usually not an issue;0464

usually the index is as small as it can be); the radicand contains no nth powers (so if there is an0469

nth power inside, as part of that radicand, we need to take the root of that and remove it from under the radical;0474

the same here with the cube root; in addition, the radicand contains no fractions (so something like this0482

is not in simplest form); and there are no radicals in the denominator (so this is also not in simplest form).0490

You can go through this checklist in your mind, when you think you are done simplifying,0498

and make sure that the expression you are working with meets these conditions.0502

OK, so I mentioned that you cannot have radicals in the denominator if something is going to be in simplest form.0508

So, getting rid of radicals in the denominator is known as rationalizing the denominator.0516

So, now we are going to talk about rationalizing denominators.0522

And we are going to start out just talking about when we are working with a monomial--when we have a radical in the denominator that is a monomial.0525

So, to eliminate radicals in the denominator, multiply the numerator and the denominator by a quantity, so that the radicand is an nth power.0532

What does this mean? 2 divided by √5xy: what I want to do is make this radicand (5xy)2,0543

because here, n equals 2; so I want to get 5xy, squared, as the radicand.0558

OK, so in order to do that, what I need to do is multiply the numerator and the denominator by the square root of 5xy.0579

This is going to give me...2 divided by the square root of 5xy, times the square root of 5xy, divided by the square root of 5xy.0589

Now, I am allowed to do that, because this is just 1; these would cancel out and give me 1, and I am allowed to multiply this by 1.0601

So, the numerator is 2 times √(5xy), divided by...looking at the product property:0612

the product property tells me that, if I multiply these two,0622

I am going to end up with √(5xy times 5xy), which equals √(5xy2).0626

So, when I have the index the same as the power that the radicand is raised to, I can just eliminate that radical sign, and then eliminate this power.0647

So, I have 2 here, really, although it is not written out, and a 2 here; so I can get rid of both of those and get rid of the radical sign.0658

So, this is going to equal 2√5xy, divided by 5xy.0667

Now, this is in simplest form, because I no longer have a radical in the denominator.0675

And I achieved that by multiplying both the numerator and the denominator by the square root of 5xy, so that I ended up with √(5xy2).0679

And I could take the square root of that to just get 5xy.0691

But you need to make sure that you multiply both the numerator and the denominator by the same term,0694

so that you are actually just really multiplying this entire thing by 1.0701

OK, if the expression in the denominator is a radical, but it is a binomial, you need to use a different technique.0708

We just talked about rationalizing a denominator when the radical was the monomial.0716

But if you have a binomial, then what you need to do is work with a conjugate.0721

So, first let's just review what a conjugate is.0727

Conjugates are the sum and difference of two terms--not even worrying about the radical sign right now.0731

It is something like a + b and a - b; but here, we are working with radicals, so it could be something like √2 + 1 and √2 - 1.0742

So, these two are conjugates--the same numbers; the same radical sign; the only difference is that this one is positive; this one is negative.0757

OK, so these radical expressions, a√b and c√d and a√b - c√d--the only difference here is in the sign.0769

These are called conjugates; so they can be used to rationalize denominators that are binomials.0782

For example, if I have 5 divided by the square root of 7 minus 3, well, √7 - 3...the conjugate of that would be √7 + 3.0792

So, these two are conjugates; this is a conjugate pair.0810

So, what I need to do is multiply 5, divided by the square root of 7, minus 3, times √7 + 3, over √7 + 3.0815

Now, since the numerator and the denominator are the same, I am really just multiplying by 1; so again, this is allowed.0832

All right, so in the numerator, this gives me...using the distributive property...5√7, plus 5 times 3, which is 15.0842

The denominator: recall that here, if you look at what I am doing, it is multiplying a sum and a difference.0861

So, if I had a + b times a - b, this is going to end up giving me a2...the outer term is -ab;0866

the inner term is positive ab; so that is going to cancel out, and then I am going to get -b2.0877

b times b is b2, but I have a negative sign in front of it.0886

So, I am going to end up with a2 - b2; and in this case, multiplying √7 - 3 and √7 + 3,0890

a equals √7 in this situation, and b equals 3; so this is going to give me0899

(√7)2 - (I am using this format) 32, which equals 5√7 + 15.0906

OK, I have (√7)2, √7 times √7; it is just going to give me 7.0920

Minus 32 (which is 9) is going to give me 5√7 + 15; 7 - 9 is -2, and I can put that negative out in front.0932

So, this is going to give me 5√7 + 15, divided by 2.0944

And now, the radical is gone from the denominator.0949

So, I have rationalized the denominator when I had a radical, and I had a situation where it was part of a binomial.0952

I had a denominator that was a binomial.0961

And I did that by multiplying both the numerator and the denominator by the conjugate of the denominator.0963

Another review of a concept you may have learned earlier on, which is adding and subtracting radicals:0973

radicals are like radicals if they have the same index and the same radicand.0980

For example, 4 times the cube root of 5x, plus 2 times the cube root of 5x:0985

recall that this is the index; here, the index is 3 and the radicand is 5x; here the index is 3 and the radicand is 5x.0997

So, I am going to use the distributive property, and I am going to say, "OK, I have the same; I can pull out this cube root of 5x."1006

And that leaves behind 4 + 2; so this becomes the cube root of 5x times 6, or 6 times the cube root of 5x.1017

All I did is add 4 and 2; and you can really just look at this as a variable, almost, with the radical.1034

If I had given you something like 4y + 2y, that equals 6y; and here, we are going to let y equal the cube root of 5x.1043

You could just look at it this way: that this whole thing is like a variable.1055

And you can add these two together, but the variables are the same,1058

because it is just saying 4 y's and 2 y's equal 6 y's, and that is the same idea here.1063

OK, subtraction: the same thing--you just have to be careful (as always, when you are working with subtraction) with the signs.1070

So, if I am subtracting something like 5 times the fourth root of 7yz, minus 3 times the fourth root of 7yz,1081

I check and see that I have the same index (which is 4) and the same radicand (7yz).1097

So, this becomes pulling out the same factor, which is the fourth root of 7yz, leaving behind 5 - 3.1103

This is going to give me the fourth root of 7yz times 2, or I am rewriting it as 2 times the fourth root of 7yz.1119

So, adding and subtracting like radicals is pretty straightforward.1131

Just make sure that you check and make sure that the index numbers are the same, and the radicands are the same, before you try to combine radicals.1135

Multiplying: with multiplying radicals, we are going to use the product property.1144

And if we are going to multiply sums or differences of radicals, we will be using the distributive property.1150

So, let's just start out with multiplying two monomials that involve radicals,1155

the fifth root of 2x3 times the fifth root of x2.1160

Well, the product property, recall, tells me that the square root of ab equals the square root of a, times the square root of b.1171

So, what I am doing down here, instead of going from left to right--I am going from right to left.1180

I already have these two split up, but the product property tells me I can combine them.1187

So, this would actually be the fifth root of 2x3, times x2.1192

Recall that, if you are multiplying exponents with a like base, then here, I can just add these exponents.1202

So, this is going to give me 2 times x5.1215

Now, I see that what I have (using the product property again) is the fifth root of 2, times the fifth root of x5,1224

which equals...well, the fifth root of x5 is just x, times the fifth root of 2.1237

So, you can see how multiplication and using the product property allowed me to actually simplify this.1243

First, I used the product property to multiply these two together.1251

Then, I used my property of exponents that says I add the exponents, since there are like bases here.1255

That gave me 2x5; and I saw that this is not in simplest form,1262

because I have a radicand that contains the nth power, the fifth power.1267

So then, I further simplified.1274

OK, so that is if you have monomials; now, for multiplying sums or differences of radicals, we need to use the distributive property.1276

For example, if I am multiplying 2 times the square root of 3x, plus the square root of 2, times the square root of x,1283

minus 2, times the square root of 5, we are going to use FOIL.1294

The first terms (multiplying the first terms, because I am multiplying two binomials--FOIL--First terms):1300

that is 2 times the square root of 3x, times the square root of x.1305

Plus the outer terms--that is 2√3x, times -2√5.1313

Inner terms are √2 times √x.1324

And finally, the last terms are √2 times -2√5.1332

OK, using the product property right here tells me that √a times √b is √ab.1339

So, I am going to apply that here to get 2 times 3x times x, plus 2 times -2...that is actually going to give me a -4;1347

so I am going to rewrite this as -4; √3x times 5...3 times 5x, plus the square root of 2x.1358

And then here, I have a -2 out in front; and then, that is the square root of 2 times 5.1373

I am doing some simplification: this equals 2 times 3x2, minus 4 times √15x, plus √2x, minus 2√10.1380

And I see here that I am not quite done yet, because I have an index of 2, and my radicand contains something to the second power.1396

So, I can pull this out, and I need to remember to use absolute value bars, because this was an even index,1404

and when I took that root of this, I ended up with an odd power, 1, so I need to use absolute values.1416

And I can't simplify any further, because I can't combine these, since they are not like radicals.1427

They are to the same powers, but none of them have the same radicands, so I can't add or subtract them.1433

So again, multiplication with sums or differences of radicals--you just use the distributive property,1439

like we have earlier on, when working with numbers or variables.1445

OK, in this first example, we are going to simplify this expression; and it is the fifth root of x8y9z5, divided by 243.1452

So, I am going to use the quotient property, because I know that this equals, according to the quotient property,1465

the fifth root of x8y9z5, all divided by the fifth root of 243.1473

So, recall that, in order to be in simplest form, a radical expression needs to have an index that is as small as possible;1484

no nth powers; no fractions under the radical; and no radicals in the denominator.1492

So, you should be familiar with these rules.1509

What we have is: when we started out, we knew it wasn't in simplest form, because I did have a fraction under that radical.1520

I then took care of that by using the quotient property.1529

I no longer have a big radical sign where I have this fraction under it; I split it up.1534

The only problem is that I now have a radical in the denominator.1539

So, this is still not in simplest form.1544

In addition, I also have some nth powers under here; when I work on this,1548

I will see that there are some terms here that are to the fifth power.1554

So, I can rewrite this as x5 times x3, because these have like bases, so I add the exponents.1561

That would give me x8 back, so I can see here that I have n = 5, and the radicand contains some fifth powers.1572

y9 would be y5 times y4, because I would add these to get 9 back; and then leave z as it is, z5.1582

243 is not totally obvious, but it turns out, if you work this out, that 3 to the fifth power is 243.1593

So, I also have a fifth power as part of the radicand in the denominator.1604

OK, I am going to use the product property to rewrite this with my fifth powers all together here:1610

x5y5z5 times what is left over (that is the fifth root of x3y5).1616

OK, what this gives me is the fifth root of these; well, these are all to the fifth power.1641

So, I simply remove the radical, get rid of the n, and get rid of this power to get x; the same with y and z.1649

Since this is odd, I don't have to worry about absolute value bars.1658

We only worry about that when the index is even.1663

And this is times the fifth root of x3; this should actually be y4 right here,1667

and y5 here and y4 here, divided by...well, this is the fifth root, and this is raised to the fifth power.1675

So, that just becomes 3; or I could rewrite this as xyz divided by 3, all that times the fifth root of xy4.1686

So, I double-check: is this in simplest form?1702

There are no nth powers in the radicand; what I have left is xy4; there are no fifth powers here.1705

There are no fractions under this radical sign; there are no radicals in the denominator.1713

So, this is in simplest form.1720

Here, we are asked to add and subtract some square roots; recall, though, that you can only add or subtract radicals1725

if they have the same index (which these all do) and the same radicand (which they don't).1734

However, they are not in simplest form yet; it is important to always simplify first.1739

The problem is that I have some perfect squares left here as part of the radicand that I could pull out.1744

So, I am going to rewrite this with the perfect squares factored out.1752

And I am going to use the product property; 24 is 4 times 6; 48 is 16 times 3, so I have a perfect square;1756

54 is 9 times 6, so that is another perfect square; and then, 75 is 25 times 3.1769

According to the product property, the square root of ab equals the square root of a times the square root of b.1777

So, I can rewrite this as √4 times √6, minus √16 times √3, plus √9 times √6, minus √25 times √3.1786

And as you get better at this, you might not need to write out every step.1802

But for now, it is a good idea, just to keep track.1806

This is going to give me 2√6, minus √16, which is 4, √3; plus √9, which is 3, √6; minus √25, which is 5, √3.1809

Now, I am looking, and I actually do have some like radicals that I can add, because they all have the same index;1825

and now I see that these two (let's rewrite it like this): 2√3 + 3√6, have the same radicand.1832

And then, I have -4√3 - 5√3.1847

In this situation, what I am going to do is add the 2 and the 3, and this is going to give me 5√6.1855

You are looking at this the same way that you would a variable: if I had 2y and 3y, it would become 5y.1865

Plus...this is going to be -4 and -5; that is going to be -9, and then √3, which is 5√6 - 9√3.1873

So, this is now in simplest form; and when I looked at this, it first looked like I could not combine these.1889

But when I went about my checklist of how to simplify, I didn't have to worry about radicals in the denominator1897

or fractions under the radical sign, but I did have to get rid of the perfect squares that were part of the radicand.1902

And I used the product property to achieve that.1909

Once I did that, I saw that I actually could combine some of these radicals to get this simplest form.1912

OK, simplify: this time, I am asked to multiply two binomials.1922

And I am going to use FOIL, just like I normally would.1927

I am rewriting this here; use FOIL just as though you are multiplying any other two binomials.1932

So, this is going to give me 2√3 times the other first term, which is 6√3,1942

plus 2√3 (the outer terms) times 2√5.1950

Now the inner terms are -4√5 times 6√3; and then the last terms are -4√5 times 2√5.1958

OK, I am making sure that I have everything correct...inner, and then last.1975

OK, we can use the product property; and the product property tells me that √ab = √a times √b.1979

And I am actually moving from right to left here, because these are separated, and I want to put them together.1992

So, this is going to give me 2 times 6, √3 times 3, plus 2 times 2, and then this is √3 times 5,1998

plus -4 times 6, and this is √5 times 3, plus -4 times 2, times √5 times 5.2015

This gives me 12; and I could rewrite this as 32 + 4; this is 15.2029

-4 times 6 is going to give me -24√15; -4 and 2 is going to give me -8; and I can write this as 52.2038

OK, I am not done simplifying yet, because recall that I look at the index; it is 2;2049

and I see that I do have a term here and here that are to the power of 2.2055

I can take 32; that square root is just going to be 3; so this gives me 12 times 3.2062

This does not have any perfect squares within it as factors, so I leave it alone.2069

The same with this term; here, I do have 52, so the square root of 5 squared is just 5.2076

Continuing to simplify: 12 times 3 is 36; -8 times 5 is -40.2085

I can combine these two, 36 - 40; that is going to give me -4.2096

I also can combine these two, because these are the same index, 2, and they are the same radicand.2106

So, this would be the same as 4 - 24 times √15,2116

which is going to give me -4; and then 4 - 24 is just going to give me -20√15.2128

So, the simplified expression here is this; and I know it is in simplest form, because I don't have any fractions under the radical sign.2140

There are no radicals in the denominator; and I don't have any nth powers here; I don't have any perfect squares under here.2151

So, I first multiplied these two binomials out, using the distributive property.2160

I got down to here; then I took my perfect squares; I took the square root of this number, 9, which is 3.2167

I took this square root of 52, which is 5.2176

I did some more simplifying, and then combined these two radicals that had like radicands and had the same index.2181

OK, Example 4: Thinking about my rules, is this is simplest form?2194

No; it doesn't have any fractions under the radical; however, there is a radical in the denominator.2200

So, a radical expression is not in simplest form if there is a radical in the denominator.2208

Recall that, to simplify a radical binomial expression, you multiply both the numerator and the denominator by the conjugate of the denominator.2214

So here, I have 2 + √3; the conjugate of that is going to be 2 - √3.2223

So, these are conjugates; this is a conjugate pair.2232

I am going to take 4 - √3, divided by 2 + √3; and I am going to multiply that times this conjugate, 2 - √3.2238

This is just the same as multiplying this by 1.2255

I am going to have to use the distributive property, because I am multiplying these binomials.2259

So here, I am just going to have to use FOIL, as usual.2266

The first two terms are going to give me 4 times 2; the outer is going to give me 4 times -√3.2269

The inner two terms--that is -√3 times 2; and then, the last two terms are -√3 times -√3.2279

The denominator is a little bit easier, because this denominator is in the form (a + b) (a - b).2294

It is the product of a sum and a difference, which gives me a2 - b2.2303

Here, a = 2, and b = √3; so that is going to give me a2, which is 22, minus (√3)2.2307

OK, simplifying: 4 times 2 is 8; this is 4 times -1, so that is -4√3; this is -1, essentially, in front of here, times 2 is -2√3.2325

Here, I have a negative and a negative; that is going to give me a positive, so it is going to be + √3; and that is squared.2346

OK, all divided by...22 is 4; minus...well, the square root of 3 squared is just 3.2356

OK, so this gives me 8 - 4√3 - 2√3.2368

Well, this √3 squared is also 3, so I am going to change that to a 3...divided by 4 - 3, which is 1...2383

so I can just not write that 1; and here I have 8 + 3; that is 11; I also see that I have -4√3 and -2√3.2391

Since these have the same index and the same radicand, I can combine these two to get -6√3.2406

So, we started out with something not in simplest form, because it had a radical in the denominator.2414

And since that was a binomial, I multiplied both the numerator and the denominator by the conjugate of the denominator.2419

I got this whole thing; I then just continued to simplify.2427

And when I got to here, I saw that I had two radicals that could be combined to give me -6√3.2434

And then, I combined my constants.2444

And this is now in simplest form, because there are no radicals in the denominator;2446

there are no nth powers in this radicand (no perfect squares, in this case);2450

and no fractions under the radical sign; and the index is the smallest power that it can be.2458

That concludes this session of Educator.com; thanks for visiting!2466

Educator®

Please sign in for full access to this lesson.

Sign-InORCreate Account

Enter your Sign-on user name and password.

Forgot password?

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
Get immediate access to our entire library.

Sign up for Educator.com

Membership Overview

  • Unlimited access to our entire library of courses.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lesson files for programming and software training practice.
  • Track your course viewing progress.
  • Download lecture slides for taking notes.

Use this form or mail us to .

For support articles click here.