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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Complex Fractions

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

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

19m 26s

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

12m 6s

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

18m 52s

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

12m 22s

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

28m 52s

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

12m 2s

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

28m 41s

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

24m 26s

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

22m 55s

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

27m 58s

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

20m 36s

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

21m 24s

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

16m 12s

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

25m 54s

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

20m 1s

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

19m 40s

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

24m 34s

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

17m 13s

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

16m 13s

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

14m 12s

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

17m 7s

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

15m 33s

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

21m 13s

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

44m 51s

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

18m 33s

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

25m 7s

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

44m 56s

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

28m 27s

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

21m 44s

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

30m 9s

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

30m 14s

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

23m 38s

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

29m 27s

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

16m 47s

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

29m 4s

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

26m 53s

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

21m 42s

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

26m 41s

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

20m 24s

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

18m 23s

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

16m 24s

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

18m 54s

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

20m 20s

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

27m 4s

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

14m 32s

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

15m 12s

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

17m 22s

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

19m 24s

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

15m 5s

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

29m 16s

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

1 answer

Last reply by: Professor Eric Smith
Sun Feb 22, 2015 4:16 PM

Post by Micheal Bingham on February 21, 2015

Hello Mr. Smith, I have a question pertaining to the third example in this video.  When you find the LCD (s^2*t^2) to multiply both fractions by, why is it that you only multiply the numerator by aforementioned LCD?  Sorry for the confusion, and thank you in advance.

Complex Fractions

  • A complex fraction is where the numerator or the denominator of a rational expression contains fractions as well.
  • To simplify a complex fraction you can combine the expressions in the numerator and denominator. Then transform it into a division problem. Remember when dividing to flip the second expression and multiply.
  • An alternate way to simplify complex fractions is to multiply the numerator and denominator by the LCD of all the small fractions. Remember to distribute if using this method.
  • Both methods for simplifying complex fractions should work, however using the LCD is often quicker and cleaner with fewer opportunities for a mistake.

Complex Fractions

Write as a rational expression:
4e2 − 3 − [(e + 1)/(e + 5)]
  • [(( e + 5 ) ×( 4e2 − 3 ))/(( e + 5 ) ×1)] − [(e + 1)/(e + 5)]
  • [(4e2 − 3e + 20e2 − 15)/(e + 5)] − [(e + 1)/(e + 5)]
  • [(4e2 − 3e + 20e2 − 15 − e − 1)/(e + 5)]
[(4e2 + 20e2 − 4e − 16)/(e + 5)]
Write as a radical expression:
7a2 − 10 + [(a − 8)/(a + 6)]
  • [(( a + 6 ) ×( 7a2 − 10 ))/(( a + 6 ) ×1)] + [(a − 8)/(a + 6)]
  • [(7a3 − 10a + 42a2 − 60 + a − 8)/(a + 6)]
[(7a3 + 42a2 − 9a − 68)/(a + 6)]
Write as a radical expression:
12b2 + 3 − [(b + 5)/(b − 1)]
  • [(( b − 1 )( 12b2 + 3 ))/(( b − 1 )( 1 ))] − [(b + 5)/(b − 1)]
[(12b3 + 3b − 12b2 − 3 − b − 5)/(b − 1)]
Simplify:
[[([(x2)/(y2)])/([(3x + 1)/(4x − 5)])]]
  • [([a/b])/([c/d])] = [ad/bc]
  • [(x2( 4x − 5 ))/(y2( 3x + 1 ))]
[(4x3 − 5x2)/(3xy2 + y2)]
Simplify:
[[([x/(y3)])/([(7x − y)/(6x + 11y)])]]
  • [(x( 6x + 11y ))/(y3( 7x − y ))]
[(6x2 + 11xy)/(7xy3 − y4)]
Simplify:
[([(m2)/n])/([(m + n2)/(2m − n2)])]
  • [(m2( 2m − n2 ))/(n( 2m − n2 ))]
[(2m3 − m2n2)/(2mn − n3)]
Simplify:
[([(x2 + 3x − 4)/(x2 + 5x + 6)])/([(x2 − 5x + 4)/(x2 − 2x − 8)])]
  • [(x2 + 3x − 4)/(x2 + 5x + 6)] ×[(x2 − 2x − 8)/(x2 − 5x + 4)]
[[(x + 4)/(x + 3)]]
Simplify:
[([(x2 + 12x + 35)/(x2 + 8x + 12)])/([(x2 − 25)/(x2 − 3x − 10)])]
  • [(x2 + 12x + 35)/(x2 + 8x + 12)] ×[(x2 − 3x − 10)/(x2 − 25)]
  • [(( x + 7 )( x + 5 ))/(( x + 2 )( x + 6 ))] ×[(( x − 5 )( x + 2 ))/(( x + 5 )( x − 5 ))]
[(x + 7)/(x + 6)]
Simplify:
[([(x2 − 64)/(x2 + 5x − 24)])/([(x2 + x − 72)/(x2 + x − 12)])]
  • [(x2 − 64)/(x2 + 5x − 24)] ×[(x2 + x − 12)/(x2 + x − 72)]
  • [(( x − 8 )( x + 8 ))/(( x + 8 )( x − 3 ))] ×[(( x − 3 )( x + 4 ))/(( x − 8 )( x + 9 ))]
[(x + 4)/(x + 9)]
Simplify:
[([(x + 6)/(x2 + 12x + 20)])/([(x + 1)/(x2 + x − 2)])]
  • [(x + 6)/(x2 + 12x + 20)] ×[(x2 + x − 2)/(x + 1)]
[(( x + 6 )( x2 + x − 2 ))/(( x2 + 12x + 20 )( x + 1 ))]

*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

Complex Fractions

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

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

Transcription: Complex Fractions

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at some complex fraction.0002

We will first have to do a little bit about explaining what a complex fraction is0012

and then I’m going to show you two techniques and how you can take care of them.0015

In the first one we will write out these complex fractions as a division problem0018

and then we will go ahead and use the method of this common denominator in order to simplify them.0023

Each of these methods has their own advantages but they both should work when dealing with simplifying complex fraction.0029

What a complex fraction is, it is either the numerator or the denominator is also a fraction.0040

Here is a good example of a complex fraction using numbers.0047

You will notice that the main division bars is actually sitting right here.0050

But in the numerator I have 2/5 and in the denominator I have 1/7.0055

If I have fractions made up of other fractions.0061

These are exactly the types of things that we are looking to simplify.0064

Since we are on a lot of rational expressions, then we will not only look at just numbers but we will look at more complex fractions like this.0072

One of the first techniques that we can use to clean this up is to use division.0082

We will use that main division bar and write this as a division problem.0089

Here I have (x + ½) ÷ (6x + 3)/4x.0095

It is the main division bar right there that will turn into division.0101

That means we will have to use all of our tools for simplifying the left and right, and eventually be able to get them together.0106

In a previous lesson we learn that we need to flip the second rational expression then multiply across the top and bottom.0115

That is exactly what you will see with these.0120

Let us grab on these rational expressions and give it a try.0125

You have (t2 u3) / r ÷ t4u/r2.0127

We want to identify what is on the top and what is on the bottom?0135

We want to write those again as a division problem.0140

(t2 u3) / r ÷ t4u/r20144

It looks pretty good.0157

We want to turn this into a multiplication process by flipping that second rational expression t2 u3/ r will now be multiplied by r2 ÷ t4u.0158

That looks much better.0177

We can go ahead and multiply across the top.0178

I will just put all of these on the top and multiply across the bottom.0182

I will put of all of these on the bottom.0187

Now that we have this we would simply go through and cancel out our common factors.0190

We will get rid of t2 on the top and t2 in the bottom.0198

Making a t2 in the bottom.0204

We can cancel out u, bring this down to u2 and we can cancel out on r.0207

We have r in the top.0217

Let us write down everything that is left over.0220

u2 r ÷ t2 and then we could consider this one simplified.0222

It changes it into a problem that we have seen before.0231

You just have to do a lot of work with simplifying.0234

When it gets to simplifying a complex fraction, even that process is not necessarily the easiest to go through.0240

In fact, you will find that this next problem is quite lengthy.0248

We have (1/x + 1 + 2/y – 2) ÷ (2 /y – 2 – 1/x + 3).0251

Let us identify everything on the top and everything in the bottom.0262

That way we can simply rewrite this 1/x + 1 + 2/y – 2 all of this is being divided by everything on the bottom 2/y – 2 – 1/x + 3.0269

If I have any hope I'm doing this as a division problem that I need to normally flip that second fraction.0298

Notice how in this one I do not have a single fraction.0305

It is tempting to say, hey why we just flip both of them but that is not how division works.0309

We need to combine it into a single fraction before we can flip it and then do the multiplication.0313

Let us see if we can get these guys together with some common denominators.0322

The common denominators on this side are the x + 1 and y – 2.0327

In order to get those together, I would have to give this fraction, y - 2 on the top and bottom.0343

To make it workout over here we will give the top and bottom of that one x +1 and the 2 is still up there.0353

It looks like that first piece will turn into y – 2 + let us go ahead and distribute this guy in there (2x + 2) ÷ (x + 1) (y – 2).0365

That just takes that and crunches it down a little bit.0384

Let us focus on this other one.0388

We need a common denominator and I see there is a y - 2 and x + 3.0390

I will need to give the fraction on the left an additional x + 3.0408

We still have the 2 in there.0412

2 × x + 30414

Over on the other side let us give the top and bottom of that one y -2.0417

When those are put together, we will do a little bit of distributing here.0424

We will have (2x + 6) – (y + 2) ÷ (y – 2) (x + 3).0431

Let us go ahead and write this again and see if we can do the actual division.0447

It looks like I can cancel out a few things in here.0455

Let us save ourselves a little bit of work.0458

y + 2x ÷ (x + 1) (y -2), we are dividing it by the second fraction.0461

Here is when I’m going to flip and multiply it, multiplied by 2x - y + 8.0476

I have combined the 6 and the 2 together, y - 2 x + 3.0487

I can just combine the tops and bottoms.0498

Quite a lot of pieces in here that is okay, at least I see one piece will cancel out and that is the y – 2.0516

We are left with y + 2x and (x + 3) ÷ (x +1) and 2x - y + 8.0528

Even though it can be a lengthy process, by rewriting it as a division process and using our tools from before0543

you will see is that it is possible to reduce and simplify this complex fractions.0549

The other method which can often be a lot cleaner is using the least common denominator0556

in order to clear out all the fractions in the top and bottom.0561

In order for this method to work, you must find the least common denominator of all the little fractions present in your complex fraction.0566

Go ahead and look at your numerator and denominator and think of all the least common denominator for all those fractions.0573

Once you find that LCD, then you are going to multiply on the top and the bottom of the main fraction0580

or on the top and bottom of the main division bar.0586

This will clear up things immensely but you have to be careful on canceling out.0591

You do not want to accidentally cancel out something that you should not.0596

You will see that you will clear out a bunch of stuff and then you end up simplifying just as you would normally.0598

No matter what method you use, you should get the same answer.0608

Use the method that you are more comfortable with.0614

I like to recommend a method two because it is usually much cleaner than using the first method.0620

However, anything that is cleaner use less opportunities for mistakes.0627

One downside to the second method is usually happen so quickly it is hard to keep track of everything that was in there.0633

Let us try this second method with the following complex fraction.0643

I have (2/s2 t + 3 /st2) ÷ (4/s2 t2 – 1/st).0647

Let us first see if we can identify the least common denominator.0656

Try and pick out all these little denominators here, see what the LCD would be.0661

They all have some s and the largest one I see there is an s2.0668

They all have t’s but the largest one in there is a t2.0675

I’m going to take this and I’m going to multiply it on the top and the bottom of this expression so I can rewrite it.0680

There is our main division bar.0698

Let us get to multiplying.0708

On the top of this entire main division bar I will multiply it by this LCD.0709

I will do the same thing on the bottom to keep things nice and balanced.0715

The top and bottom both have two terms, so we will definitely make sure that we distribute each of these parts.0719

We are going to write the result of this multiplication and then watch how many things will cancel out in the next step.0730

(2s2 t2 / s2 t) + (3s2 t2 / st2) ÷ (4s2 t / s2 t2 - s2 t/st).0737

It looks messy and it looks I have actually made things even more complicated.0768

Watch what is going to go take a vacation here soon.0772

I have these s2 they will go away.0776

Then I have a single t and t up top, they will go away.0779

On the next one, there is my s one of those will be gone and both my t2 are gone.0783

Onto the bottom s2 is gone, s2 is gone.0790

I think I am missing some of my squares.0797

Our t2 are gone and we can get rid of one of these t’s.0805

The denominators of all those little fractions, these guys that we are so worried about at the very beginning,0817

all of them have been cancel out in some way or another.0823

This means as soon as we write down our leftover pieces, this one is simplified.0826

2t + 3s, those are the only things that survived up here ÷ 4 - st and those are the only things that survived on the bottom.0831

This one is in its most simplified form.0847

It is nice, quick and easy method you just have to properly identify the LCD first.0852

You will know you are using the method right if all of these denominators end up going away.0858

If any of them are still in there double check to see what LCD you used.0862

Let us try this one more time but something a little bit more complicated.0869

This one is (2y + 3 / y – 4) ÷ (4y2 – 9) + (y2 – 16).0873

Let us examine these denominators so we can find our LCD.0882

We are looking at this one this is the same as y - 4 and y + 4.0888

Over here it already has the y -4 in it.0897

The only piece that I am missing is the y + 4.0900

My LCD will contain both of these parts.0905

I have the y – 4 and y + 4, both of those in there.0908

I’m going to take that and we are going to multiply it on the top and bottom of the original.0917

Let me just quickly rewrite this and I’m going to rewrite it with the factored form on the bottom.0923

We will take our LCD and we will multiply it on the top and on the bottom of our main fraction here.0939

y – 4 y + 4.0952

Let us go ahead and put everything together and let us see how this looks.0959

I have (2y + 3 ) (y – 4) (y + 4) ÷ y – 4.0965

Then comes our main division bar right there.0980

On the bottom is (4y2 – 9) × (y – 4) × (y + 4) ÷ (y – 4) (y + 4).0985

Watch how many things will cancel in this next step.1005

y – 4 and y – 4 those are gone.1009

y – 4, y – 4, y + 4, y + 4, 4 those are gone.1013

All of these problems that we had at the very beginning, they are no longer problem.1017

They are gone.1022

We will simply write down all of the left over pieces.1023

2y + 3 y + 4 4y2 – 9.1028

Be careful, there is still some additional reducing that you can do even after using your LCD like this.1044

One thing that I can see is that I can actually continue factoring the bottom.1050

Let us write that out.1056

2y + 3/ y + 4 and this will be over I have different squares on the bottom so, 2y + 3, 2y – 3.1058

I’m sure enough now we can more easily that I have an extra 2y – 3 in the bottom and that is gone as well.1074

This one finally reduces down to y + 4 / 2y -3 and now we are finally done.1081

The second method is definitely handy and clears up a lot of fractions very quickly.1092

Use whatever method you are more comfortable with.1097

Thank you so much for watching www.educator.com.1101