Dr. Carleen Eaton

Operations with Radical Expressions

Slide Duration:

Table of Contents

Section 1: 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

Section 2: 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

Section 3: 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

Section 4: 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

Section 5: 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

Section 6: 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

Section 7: 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

Section 8: 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

Section 9: 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

Section 10: 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

Section 11: 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

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Algebra 2 Operations with Radical Expressions

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Section 7: Radical Expressions and Inequalities: Lecture 5 | 41:11 min

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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

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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√6 3*3√2
− 6√6 9√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

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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

Algebra 2

Section 1: Equations and Inequalities
	Expressions and Formulas	22:23
	Properties of Real Numbers	20:15
	Solving Equations	19:10
	Solving Absolute Value Equations	17:31
	Solving Inequalities	17:14
	Solving Compound and Absolute Value Inequalities	25:00
Section 2: Linear Relations and Functions
	Relations and Functions	32:05
	Linear Equations	14:46
	Slope	23:07
	Writing Linear Functions	23:05
	Special Functions	31:05
	Graphing Inequalities	21:42
Section 3: Systems of Equations and Inequalities
	Solving Systems of Equations by Graphing	17:13
	Solving Systems of Equations Algebraically	23:53
	Solving Systems of Inequalities By Graphing	27:12
	Solving Systems of Equations in Three Variables	28:53
Section 4: Matrices
	Basic Matrix Concepts	11:34
	Matrix Operations	21:36
	Matrix Multiplication	29:36
	Determinants	33:13
	Cramer's Rule	28:25
	Identity and Inverse Matrices	22:25
	Solving Systems of Equations Using Matrices	22:32
Section 5: Quadratic Functions and Inequalities
	Graphing Quadratic Functions	31:48
	Solving Quadratic Equations by Graphing	27:03
	Solving Quadratic Equations by Factoring	19:53
	Imaginary and Complex Numbers	35:45
	Completing the Square	27:11
	Quadratic Formula and the Discriminant	22:48
	Analyzing the Graphs of Quadratic Functions	30:07
	Graphing and Solving Quadratic Inequalities	27:05
Section 6: Polynomial Functions
	Properties of Exponents	19:29
	Operations on Polynomials	13:27
	Dividing Polynomials	31:11
	Polynomial Functions	22:30
	Analyzing Graphs of Polynomial Functions	33:29
	Solving Polynomial Functions	21:10
	Remainder and Factor Theorems	31:21
	Roots and Zeros	31:27
	Rational Zero Theorem	31:16
Section 7: Radical Expressions and Inequalities
	Operations on Functions	34:30
	Inverse Functions and Relations	22:42
	Square Root Functions and Inequalities	30:04
	nth Roots	20:46
	Operations with Radical Expressions	41:11
	Rational Exponents	30:45
	Solving Radical Equations and Inequalities	31:27
Section 8: Rational Equations and Inequalities
	Multiplying and Dividing Rational Expressions	40:54
	Adding and Subtracting Rational Expressions	55:04
	Graphing Rational Functions	57:13
	Direct, Joint, and Inverse Variation	20:21
	Solving Rational Equations and Inequalities	55:14
Section 9: Exponential and Logarithmic Relations
	Exponential Functions	35:58
	Logarithms and Logarithmic Functions	45:54
	Properties of Logarithms	28:43
	Common Logarithms	25:23
	Base e and Natural Logarithms	21:14
	Exponential Growth and Decay	24:30
Section 10: Conic Sections
	Midpoint and Distance Formulas	32:42
	Parabolas	41:27
	Circles	21:03
	Ellipses	46:51
	Hyperbolas	38:15
	Conic Sections	18:43
	Solving Quadratic Systems	47:04
Section 11: Sequences and Series
	Arithmetic Sequences	21:16
	Arithmetic Series	21:36
	Geometric Sequences	23:03
	Geometric Series	22:43
	Infinite Geometric Series	18:32
	Recursion and Special Sequences	14:34
	Binomial Theorem	48:30

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 n^th 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 n^th 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 n^th 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 n^th 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 n^th powers (so if there is an0469

n^th 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 n^th 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 n^th 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 n^th 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 n^th 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 n^th 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 n^th 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 n^th 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

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Algebra 2 Operations with Radical Expressions

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	− 2√{54}	3√{18}
Reduced Roots	− 2*3√6	3*3√2
	− 6√6	9√2

	− 3√{45}	− 2√{20}
Reduced Roots	− 3*3√5	− 2*2√5
	− 9√5	− 4√5

Algebra 2 Operations with Radical Expressions

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