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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Rational Exponents

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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
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Lecture Comments (5)

2 answers

Last reply by: Kyoung-Hee Kim
Tue Oct 7, 2014 7:47 PM

Post by Kavita Agrawal on June 19, 2013

I don't think Example 1 is completely simplified. The 5th root of 512c^6 can be written as c times the 5th root of 512c, because c^6 = c^5 * c.

1 answer

Last reply by: Dr Carleen Eaton
Thu May 24, 2012 8:12 PM

Post by Darren Fuller on May 15, 2012

How would I solve a problem like this

2^5/2 - 2^3/2

Rational Exponents

  • All the properties of integer valued exponents remain true for rational exponents.
  • In simplified form, all exponents must be positive and exponents in the denominator must be integers.

Rational Exponents

Write as a radical (9x3)[4/7]
  • Recall the formula a[m/n] = n√{am}
  • Identify
  • a =
  • m =
  • n =
  • a = 9x3
  • m = 4
  • n = 7
  • Plug - in the values
  • (9x3)[4/7] = n√{am} = 7√{( 9x3 )4}
  • Simplify
7√{( 9x3 )4} = 7√{6561x12}
Write as a radical (2x4)[2/3]
  • Recall the formula a[m/n] = n√{am}
  • Identify
  • a =
  • m =
  • n =
  • a = 2x4
  • m = 2
  • n = 3
  • Plug - in the values
  • (2x4)[2/3] = n√{am} = 3√{( 2x4 )2}
  • Simplify
3√{( 2x4 )2} = 3√{4x8}
Write as a radical (5x7)[4/5]
  • Recall the formula a[m/n] = n√{am}
  • Identify
  • a =
  • m =
  • n =
  • a = 5x7
  • m = 4
  • n = 5
  • Plug - in the values
  • (5x7)[4/5] = n√{am} = 5√{( 5x7 )4}
  • Simplify
5√{( 5x7 )4} = 5√{625x28}
Write with rational exponents a radical 5√{(625 − x2)28}
  • Recall the formula a[m/n] = n√{am}
  • Identify
  • a =
  • m =
  • n =
  • a = 625 − x2
  • m = 28
  • n = 5
  • Plug - in the values
5√{(625 − x2)28} = a[m/n] = (625 − x2)[28/5]
Write with rational exponents a radical 4√{(x2 − 81)3}
  • Recall the formula a[m/n] = n√{am}
  • Identify
  • a =
  • m =
  • n =
  • a = x2 − 81
  • m = 3
  • n = 4
  • Plug - in the values
4√{(x2 − 81)3} = a[m/n] = (x2 − 81)[3/4]
Write with rational exponents a radical 5√{(x2 + x + 5)2}
  • Recall the formula a[m/n] = n√{am}
  • Identify
  • a =
  • m =
  • n =
  • a = x2 + x + 5
  • m = 2
  • n = 5
  • Plug - in the values
5√{(x2 + x + 5)2} = a[m/n] = (x2 + x + 5)[2/5]
Simplify [(x[3/4])/(x[7/8])]
  • Recall that the rules of simplfying, which are:
  • 1. No negative exponents
  • 2. No fractional exponents in denominator
  • 3. Not a complex fraction
  • 4. Index as small as possible
  • We are going to use these rules to guide us through the problem.
  • Step 1: Using rules of exponents, simplify remember [(am)/(an)] = am − n
  • [(x[3/4])/(x[7/8])] = x[3/4] − [7/8]
  • In order to subtract the fraction, we need to have the same denominator, in this case 8
  • x[3/4] − [7/8] = x[6/8] − [7/8] = x − [1/8]
  • Eliminate negative exponents using the rule a − n = [1/(an)]
  • x − [1/8] = [1/(x[1/8])]
  • We need to eliminate the fractional exponent in the denominator.
  • We need to multiply by a number (x?) such that when multiplied with (x[1/8]) equals x
  • Multiply numerator and denominator by (x[7/8])
  • [1/(x[1/8])]*( [(x[7/8])/(x[7/8])] ) = [(x[7/8])/x]
All rules of simplifying are met, no further simplifying is needed.
Simplify [(x[1/5])/(x[1/2])]
  • Recall that the rules of simplfying, which are:
  • 1. No negative exponents
  • 2. No fractional exponents in denominator
  • 3. Not a complex fraction
  • 4. Index as small as possible
  • We are going to use these rules to guide us through the problem.
  • Step 1: Using rules of exponents, simplify remember [(am)/(an)] = am − n
  • [(x[1/5])/(x[1/2])] = x[1/5] − [1/2]
  • In order to subtract the fraction, we need to have the same denominator, in this case 10
  • x[1/5] − [1/2] = x[2/10] − [5/10] = x − [3/10]
  • Eliminate negative exponents using the rule a − n = [1/(an)]
  • x − [3/10] = [1/(x[3/10])]
  • We need to eliminate the fractional exponent in the denominator.
  • We need to multiply by a number (x?) such that when multiplied with (x[3/10]) equals x
  • Multiply numerator and denominator by (x[7/10])
  • [1/(x[3/10])]*( [(x[7/10])/(x[7/10])] ) = [(x[3/10])/x]
All rules of simplifying are met, no further simplifying is needed.
Simplify [(x[2/3])/(x[3/4])]
  • Recall that the rules of simplfying, which are:
  • 1. No negative exponents
  • 2. No fractional exponents in denominator
  • 3. Not a complex fraction
  • 4. Index as small as possible
  • We are going to use these rules to guide us through the problem.
  • Step 1: Using rules of exponents, simplify remember [(am)/(an)] = am − n
  • [(x[2/3])/(x[3/4])] = x[2/3] − [3/4]
  • In order to subtract the fraction, we need to have the same denominator, in this case 12
  • x[2/3] − [3/4] = x[8/12] − [9/12] = x − [1/12]
  • Eliminate negative exponents using the rule a − n = [1/(an)]
  • x − [1/12] = [1/(x[1/12])]
  • We need to eliminate the fractional exponent in the denominator.
  • We need to multiply by a number (x?) such that when multiplied with (x[1/12]) equals x
  • Multiply numerator and denominator by (x[11/12])
  • [1/(x[1/12])]*( [(x[11/12])/(x[11/12])] ) = [(x[11/12])/x]
All rules of simplifying are met, no further simplifying is needed.
Simplify [(x[2/3])/(x[1/2] − 3)]
  • Recall that the rules of simplfying, which are:
  • 1. No negative exponents
  • 2. No fractional exponents in denominator
  • 3. Not a complex fraction
  • 4. Index as small as possible
  • We are going to use these rules to guide us through the problem.
  • Step 1: Multiply the numerator and denominator by the cojugate of the denominator. Use the shorcut a2 − b2.
  • [(x[2/3])/(x[1/2] − 3)]*[(x[1/2] + 3)/(x[1/2] + 3)] = [(x[2/3](x[1/2] + 3))/((x[1/2])2 − (3)2)]
  • Step 2: Simplify as much as possible
  • [(x[2/3](x[1/2] + 3))/((x[1/2])2 − (3)2)] = [(x[2/6] + x[2/3])/(x − 9)] = [(x[1/3] + x[2/3])/(x − 9)]
Notice all rules of simplifying are met, so you're done.

*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

Rational Exponents

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
  • Definition 1 0:20
    • Example: Using Numbers
    • Example: Non-Negative
    • Example: Odd
  • Definition 2 4:32
    • Restriction
    • Example: Relate to Definition 1
    • Example: m Not 1
  • Simplifying Expressions 7:53
    • Multiplication
    • Division
    • Multiply Exponents
    • Raised Power
    • Zero Power
    • Negative Power
  • Simplified Form 13:52
    • Complex Fraction
    • Negative Exponents
    • Example: More Complicated
  • 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

Transcription: Rational Exponents

Welcome to Educator.com.0000

Today, we will be talking about rational exponents.0002

And this is a concept that may be new, so let's start out by talking about what rational exponents are.0005

They are actually exponents that are fractions; so you may also hear these called fractional exponents.0012

OK, starting out with a definition: for a positive integer n, a1/n = the nth root of a.0020

We will talk about this restriction in a second: first, let's just look at what this is saying, using numbers.0034

So, for a positive integer n, a1/n equals this.0040

So, let's let a equal 5 and n equal 2; if a equals 5 and n equals 2, then this is going to give me 51/2, to the one-half power.0047

This equals the square root of 5, since we don't actually write the 2.0065

When you see this notation, what you need to do is take this number to the whatever root this denominator is.0072

This...we have talked before that n, right here by this radical, is the index.0084

So, in this case, we put the index right here: 1 over the index; it is the number to 1 over the index.0091

You may also see this with other values of n, and with variables involved.0100

3x to the 1/3 power equals...well, this is the index, so I am going to put it right here...the cube root of this 3x.0106

I could proceed in the other direction: I may have something that is saying "the sixth root of a."0116

And I may be asked to write it using a rational exponent or a fractional exponent--an exponent that is a fraction.0124

OK, so I write the a here; and I do 1 over the index; the index here is 6, n = 6, so this is 81/6.0132

Now, let's look at this restriction: and we have seen similar restrictions when we have even indices in other situations.0144

If n is even, then a must be greater than or equal to 0.0152

a must be a non-negative number when n is even: let's think about why.0159

If I have something such as -4, and n is 2; so I am letting a equal -4 and n equal 2; that would be written as such.0166

Well, this equals the square root of -4, and my index is 2; so it is just written as a square root.0181

This is going to end up giving me something that is not a real number; and we are just staying within real numbers,0190

restricting our discussion to real numbers, although we have learned about imaginary and complex numbers in this course.0195

Therefore, we are not allowing that; we are going to have this restriction that, if n is even, then a must be greater than or equal to 0.0201

I don't run into that problem if n is odd; and the reason is--let's say n is 3, and I am going to have a equal -8.0210

a is allowed to equal a negative number if n is odd.0221

This is going to be -81/3; -8 goes here, and this is the index; so it is the cube root of -8.0225

Recall that -2 cubed would be -2 times -2 times -2, which would be -8.0237

Therefore, the cube root of this negative number is a real number, whereas an even root of a negative number is not going to give you a real number.0252

That is why this restriction is only for when n is even.0264

A more general case: for positive integers m and n, am/n equals the n root of am.0272

Or we can also rewrite this as the nth root of a, raised to the m power.0282

Again, we have the restriction that, if n is even, then a must be greater than or equal to 0.0292

In the last case we saw, that was just a special case of this where m was 1.0298

So, before, we said that, if we had something like 41/2, I could just say, "OK, m equals 1, a equals 4, and n equals 2."0304

And then, I am rewriting that as the square root (because the index is 2) of 4.0317

And m was 1, so I didn't actually write it; I don't write all this out in this case--I just say the square root of 4.0322

Now, we are talking about situations where m is something other than 1, but it is the same basic concept.0331

For example, if I have 23/4, here m equals 3; a equals 2; and n equals 4.0338

So then, I am going to have 2 under the radical; this is going to be the fourth root of 23.0350

With variables, you could have something like x3/2, and that is going to give me x under the radical;0360

the index is 2; and the power that x is raised to is 3.0370

Again, we have this restriction that if n is even, right here, then a must be greater than or equal to 0.0376

So, even if m is something other...for example, m could be 7/2...this would be...3 goes into the radical;0386

this is the index, n, and then m is to the seventh power.0411

So, since this is even, I do have the restriction that whatever is under here has to be positive.0414

So, if I had a variable, this would give me x13; that is the fourth root of x to the thirteenth power.0419

I have the restriction here that x has to be greater than or equal to 0.0435

OK, so again, you look at what you have; you put this number under the radical.0444

The denominator is the index; the numerator is the power that the radicand is raised to.0451

And for even indexes, you need to have the restriction that the radicand must be greater than or equal to 0.0458

We talked, in the last lecture, about simplifying radical expressions.0474

And we are going to talk in more depth now; and we are going to start out by just reviewing some properties0480

that we talked about for when we are working with powers.0487

And in earlier lessons, in Algebra I, you learned these properties; and we are going to review them now,0491

because we are going to apply them to the situation where we have rational exponents.0498

So, you learn these properties for when the exponents are integers; but they still hold up when you have fractions as the exponents.0503

For example, multiplication: am times an equals am + n.0512

And we did that working with things like x3 times x2.0525

But this also applies where we might have something such as x1/2 times x3/4.0532

So, this is just going to equal x to the 1/2 + 3/4; so adding the exponents, this is going to be just 2/4 + 3/4, so this is going to be x5/4.0540

It is the same property, just using fractions up here for m and n.0562

OK, when you are working with quotients (with division), am/an equals a, and then you subtract:0567

You take m and subtract the exponent in the denominator; again, this holds true with division, if you have something with a rational exponent.0579

So, if you have something like y2/3, divided by y1/3, I just do the same thing.0589

y2/3 - 1/3 = y1/3.0601

Recall that am, raised to the n power, is am x n; you multiply the exponents.0609

The same thing here--we might have something like wz to the 1/6th is the same as w1/6 times z1/6.0618

Oh, excuse me, that is the next one.0633

You might have something such as x1/2 raised to the 3/4 power, which is going to give you x to the 1/2 times 3/4.0639

And this is going to equal x to the 3 over 2(4), so that is x3/8.0655

Now, when you have something like ab, and all of that is raised to a power, that gave you am times bm.0664

Here, what I started to show you before was (wz)1/6 = w1/6z1/6.0676

Now, another property to recall is that a (any number) to the 0 power equals 1.0688

But recall that a does not equal 0, because 0 to the 0 power is not defined.0699

And finally, something that we are going to be using again to simplify, which we used0709

in earlier lessons, is that a-n = 1/an; and this also applies when you are working with fractions.0712

So, if I had something such as a-1/5, this equals 1/a1/5.0721

Now, this first states that the properties of powers for integer exponents that we learned are valid for rational exponents.0735

And that is what I demonstrated here.0743

A couple more rules about simplifying: simplified expressions--we talked earlier on about what a simplified expression looks like.0746

when working with radicals--that we can't have radicals in the denominator; that we cannot have fraction beneath the radical...0757

Well, in addition, you will recall that simplified expressions contain only positive exponents,0766

and that exponents in the denominator must be positive integers.0772

So, if you have something like xy to the -2, that is not simplified.0778

Also, if I have something like 2x/y1/2, this is also not simplified.0789

So, wherever the exponents are (numerator or denominator), they need to be positive in order to have something be in simplest form.0798

For the denominator, in addition to saying that all of the exponents must be only positive,0807

we also say that we cannot have fractional (rational) exponents in the denominator.0813

Exponents in the denominator must be integers; so these are not simplified.0819

Let's talk now about simplifying and sum up what we have learned so far.0830

An expression with rational exponents is simplified if it has no negative exponents, there are no rational0836

or fractional exponents in the denominator, and it is not a complex fraction.0843

We just covered these two a bit; just recalling what we are talking about, you can't have a complex fraction0848

So, if you have something like 2 divided by x, over y4, divided by 4, a complex fraction--that is obviously not in simplest form.0856

And the index needs to be as small as possible.0867

Again, we are not going to be working with this too much; but it is just something to be aware of.0871

Let's focus on these two: first, not having negative exponents.0875

Since a-n equals 1/an, we can use this property in order to get rid of negative exponents.0880

So, if I have something such as 2x-3, I can simplify this by saying,0888

"OK, I will put x in the denominator, and then I can change this number to a positive."0893

Because a-1 equals 1/an, that means that x-3 is the same as 1/x3.0904

Let's look at a little bit more complicated situation, though.0915

Let's look at x-3/4: well, looking at this property, I see how to get rid of the negative--that I just rewrite it as 1/x3/4.0918

So, I am looking through, and I said, "OK, I have no negative exponents; great!"0934

But then, I look at this next rule: an expression is simplified if it has no fractional exponents in the denominator.0937

So, even though now I have gotten rid of that negative exponent, I have created a new problem.0945

And that is that I have a fractional exponent in the denominator.0950

So, let's think about how to get rid of that.0954

The way we are going to get rid of that is: I want to change this 3/4 to 1.0956

That way, this will just become x--let's think about how I can do that.0961

x (or I am going to write it out as x1 in this case) equals x3/40967

times x to something--we are just going to call that other exponent y.0976

Well, since multiplication involves adding exponents, what I am really saying is this: x1 = x3/4 + y.0982

So, I need to figure out what y is; and I know, in this one, it is pretty simple just from looking at it.0996

But this is a technique you can use with more complex problems.1000

1 = 3/4 + y; solving for y, y = 1/4; so what this tells me is that x1 = x3/4 times x1/4,1003

which you probably just figured out from looking; but again, this is a technique that you might need to use.1023

Therefore, I know that what I can do is multiply this times x1/4.1029

And if I do, it is going to give me an x in the denominator.1043

However, I need to also multiply the numerator by x1/4, because that way, this divided by this is just 1.1047

So I am really just multiplying by 1.1055

So, this is going to give me x1/4 over x; now, I have no negative exponents;1058

I have no fractional exponents in the denominator; and this is not a complex fraction; and the index is as small as possible.1068

I could even...the index here is 4; something else I could do, just to check1077

and make sure that what I ended up with is equivalent to what I started with--1083

I could use my quotient property and say, "OK, this is equal to x1/4 - 1."1087

Division would be x1/4 minus the power down here--x to the 1/4 minus 1 equals x to the -3/4.1102

So, I see that I got this back; therefore, this is equivalent to what I started out with, but it is written in simplified form, because it meets these criteria.1114

So again, the main new technique to learn (because this is review) is that,1125

if you have a fractional exponent in the denominator, you want to multiply by something1129

that will make this exponent 1, to get rid of that fractional exponent.1137

So, let's work out some examples.1144

Example 1 asks us to write this expression as a radical: (8c2)3/5.1147

Recall that am/n = n√am.1155

Here, a equals 8c2; it looks complicated, but it is just my a; m is 3, and n is 5.1167

So, right here, I am going to have 8c2 under the radical; m is 3, so I am going to raise this to 3.1181

The index is 5, so I am going to take the fifth power of that.1193

I could leave it like this, or I could go one step further and say, "OK, if you worked out 8 to the fifth power, you would get that it is 512."1197

And this is c squared to the third power; recall that, if I am going to raise a power to a power, I am going to multiply the exponents.1207

So, it is c6; now this is in the simplest form that I can get it in, because this is equivalent to 83 times (c2)3.1221

We were using this rule that shows us how to convert a fractional exponent into a radical.1231

Example 2: now we are given a radical and asked to write it with rational exponents.1241

So again, I am going to think about my definition that am/n = a...index is n, and it is raised to the m power.1247

So here, what I have is a = 3x2 - 2; m is right here--m is 4; and n is 5.1261

With that in mind, I can rewrite this as 3x...actually, we are working the other way, so I am going to rewrite this as--1273

no radical--3x2 - 2, and here I have m, so m becomes the numerator, and that is 4; and the denominator is 5.1287

So, this expression, the fifth root of 3x2 - 2 to the fourth, equals 3x2 - 2 to the 4/5.1307

All I needed to do is eliminate the radical, and then raise this to the 4/5 power.1319

And these are equivalent expressions.1326

Example 3: I am asked to simplify, and I am going to think about my rules of simplifying.1330

It is that an expression is in simplest form if it has no negative exponents;1335

if it has no fractional exponents in the denominator; if it is not a complex fraction;1345

and if the index is as small as possible.1376

So, I see here that I don't have any negative exponents right now, but I do have some fractional exponents.1380

So, what I am working with is division; so I am just going to start out by dividing.1388

This x1/2 divided by x2/3 follows the property of exponents, when we are working with quotients,1393

that tells me that I take x to the 1/2 - 2/3.1400

And this equals x to the...we are looking for the common denominator of 6, so this is x to the 3/6 - x to the 4/6.1407

So, this equals x-1/6.1422

Now, I got rid of the fractional exponent in the denominator; but now what I have is a negative exponent, x-1/6.1427

To get rid of that, I am going to use my rule that tells me that, if I have a negative exponent, I can convert it to this: 1/x1/6.1438

Now, I have something simpler-looking that I can work with.1452

I am back to having this fractional exponent in the denominator, but this is simpler to work with.1454

OK, so recall that the way we get rid of a fractional exponent is: we want to convert this to the exponent of 1 instead.1460

So, I want x1, and I have x1/6, so I need to multiply this by x to something,1468

which is the same as x1/6 + y.1477

So, I am going to set 1 equal to 1/6 + y; this gives me 1 - 1/6 = y, so y equals 5/6.1484

So, what this tells me is that I need to multiply this times x1/6.1494

Only, I have to do the same thing to the numerator and the denominator, so I am going to also multiply1507

the numerator by x5/6, so that really, I am just multiplying by 1, which is allowable.1513

And this is going to give me x5/6, and x1/6 + 5/6; that is just x.1519

So, I end up with x5/6/x; this may not look that much simpler than what we started out with,1531

but it meets the criteria for simplest form, because I now have no negative exponents;1537

I have no fractional exponents in the denominator; and this is not a complex fraction.1542

So, this is in simplest form.1548

I started out by just using my regular rule for dividing with exponents, which means that I am going to subtract the exponents.1550

Since there are like bases, I just subtract the exponents.1557

I have -1/6; I use my rule that tells me I can rewrite this as 1/x1/6 to get rid of the negative.1560

And then, I used the technique of multiplying the numerator and the denominator by x5/6,1569

because that will make this x; and I am not worried about having a fractional exponent in the numerator, because that is allowed.1575

In this fourth example, we have x5/2 over x1/2 - 2.1582

Now, recall what x1/2 is: it is actually just the square root of x.1588

So, what we have here is really the square root of x, minus 2.1594

And recall that, if there is a radical in the denominator, or a fractional exponent in the denominator,1599

which is really the same thing, then it is not in simplest form.1605

And in an earlier lesson, we talked about how, if you have a radical in the denominator that is part of a binomial,1608

you multiply both the numerator and the denominator by the conjugate, in order to eliminate that radical.1614

So, it is the same idea here, only we are working with it in this form: x1/2 - 2.1622

The conjugate is going to be x1/2 + 2, which is the same as up here, if I were to say that the conjugate would be √x + 2--same idea.1630

I need to multiply both the numerator and the denominator by that.1649

And the reason that I have to multiply both the numerator and the denominator is because,1660

that way, I am really just multiplying by 1, which is allowable.1664

This is really just multiplying by 1.1670

Using the distributive property in the top: x5/2, times this first term, x1/2, plus x5/2, times 2.1672

In the denominator, what we can do is use the fact that, when we are multiplying conjugates like this,1690

x1/2 - 2 and x1/2 + 2, it is multiplying a sum and a difference.1697

So, it is in this form: here we have the difference first, so I will write it that way.1705

But you will end up with something in this form: a2 - b2.1709

In this case, here, a is equal to x1/2, and b is equal to 2.1713

Therefore, what I am going to end up with is (x1/2)2 - 22.1720

So, I am going to put that right down here: (x1/2)2 - 22.1727

OK, since multiplying with like bases, when you have an exponent, just means to add the exponents; I am going to do the same thing here.1736

This really is just x5/2 + 1/2; so, 5/2 + 1/2 is just going to give me 6/2.1745

Plus...I am going to rewrite this as 2 times x5/2.1758

I go to the denominator, and that is going to give me (x1/2)2.1765

Well, using the rules of raising a power to a power, this is going to be x1/2 x 2, which is going to be x2/2, or x1.1771

So, it is just x; 2 squared is 4; this is going to give me 6 divided by 2, which is x3, plus 2x5/2, divided by x - 4.1788

So, it is still a pretty complicated-looking expression.1806

However, when I look, it is in simplified form.1810

I no longer have a fractional exponent in the denominator; and in order to get rid of that, it is the same concept1813

as getting rid of a radical in the denominator that is part of a binomial, because that is actually what I really have.1820

It is just written with a different notation.1826

I multiplied the numerator and the denominator by the conjugate.1828

That got rid of this fractional exponent in the denominator, and then it just came down to doing some simplifying.1832

That concludes this session of Educator.com; thanks for visiting!1841

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