# Sect Least Common Denominator

Size: px
Start display at page:

Download "Sect Least Common Denominator"

Transcription

1 4 Sect.3 - Least Common Denominator Concept #1 Writing Equivalent Rational Expressions Two fractions are equivalent if they are equal. In other words, they are equivalent if they both reduce to the same fraction. For instance, the fractions 30 1 and 0 3 are equivalent because they both reduce to 10. We can also build fractions so that they have a given denominator by multiplying the top and bottom by the same non-zero number. If we want to write 10 so that it has a denominator of 3, we would multiply top and bottom by since goes into 3 five times: (1) We can do the same thing with rational expressions. Write the following as an equivalent rational expression with the indicated denominator: Ex. 1 Ex. Ex. 3 3 xy x yz Take the new denominator divide by the old denominator: x yz xy xz. So, we need to multiply top and bottom by xz: 3 xy 3 xy (1) 3 xy xz xz 1xz x yz x 3y 1y 4 Since 1y 4 3y y 3, we need to multiply top and bottom by y 3 : x 3y x 3y (1) x 3y y3 y 3 w 10w 31w+1 49xy 3 1y 4 Since 10w 31w + 1 (w )(w 3) and

2 (w )(w 3) (w ) (w 3), then we need to multiply top and bottom by (w 3): w (w 3) (w 3) (w 3) (w )(w 3) 10w 6 (w )(w 3) Notice that we left the denominator in factored form. In general, we usually leave the denominator of a rational expression in factored form. Ex x 16 81x x 3 19x 6 9x 16 (3x 4)(3x + 4) 81x x 3 19x 6 (81x x 3 ) + ( 19x 6) x 3 (3x + 4) 64(3x + 4) (3x + 4)(x 3 64) (3x + 4)(3x 4)(9x + 1x + 16) Since (3x + 4)(3x 4)(9x + 1x + 16) (3x 4)(3x + 4) (9x + 1x + 16), then we need to multiply top and bottom by (9x + 1x + 16): 8 9x 16 8 (3x 4)(3x+4) (9x +1x+16) (9x +1x+16) x +96x+18 (3x 4)(3x+4)(9x +1x+16) 8(9x +1x+16) (3x 4)(3x+4)(9x +1x+16) Concept # Least Common Denominator When adding two fractions with different denominators, we had to first find the least common denominator. To find the LCD of two fractions, we first find the prime factorization of each denominator. Next, we write down the product of each factor that appears in the prime factorizations. Finally, we choose the highest power of the factor in the prime factorizations. The resulting product is our L.C.D. For instance, if our two denominators are 18 and 60, the prime factorization of each is: The factors, 3, and appear in the prime factorizations so we write their product: 3

3 But the highest power of the factors of is and the highest power of the factors of 3 is, so we write: 3 The principle is the same for finding the L.C.D. of rational expressions: To Find the L.C.D. of Two or More Rational Expressions: 1. Factor the denominator of each rational expression.. Write down as a product each factor that appears in at least one of the factored denominators. 3. Raise each factor in step # to the highest power of that factor that appears in the factored denominators. Find the L.C.D. of the following rational expression: Ex. Ex. 6 1, 1 a 3 b 8ab c We do not need to factor the denominators since they each have one term. The least common multiple of and 8 is 8. For the variable part, we write the product of each factor that appears: 8abc The highest power of a is 3, of b is, and c is 1. Thus, L.C.D. is 8a 3 b c x, x 4x 4 18x +1x 6 4x 4 4(x 1) 4(x 1)(x + 1) 18x + 1x 6 6(3x + x 1) 6(3x 1)(x + 1) The least common multiple of 4 and 6 is 1. For the variable part, we write the product of each factor that appears: 1(x 1)(x + 1)(3x 1) The highest power of each factor is one. Thus, L.C.D. 1(x 1)(x + 1)(3x 1) 6

4 Ex. Ex. 8, 1a 3 +48a +48a 4a +0a+4 1a a + 48a 1a(a + 4a + 4) 1a(a + ) 4a + 0a + 4 4(a + a + 6) 4(a + 3)(a + ) The least common multiple of 4 and 1 is 1. For the variable part, we write the product of each factor that appears: 1a(a + )(a + 3) The highest power of each factor is one except of the factor of (a + ); its highest power is Thus, L.C.D. 1a(a + ) (a + 3) y, 41y 1000y y 3 300y 90y y 3 + (10y) 3 + (3) 3 (10y + 3)(100y 30y + 9) 1000y 3 300y 90y + (1000y 3 300y ) + ( 90y + ) 100y (10y 3) 9(10y 3) (10y 3)(100y 9) (10y 3)(10y 3)(10y + 3) (10y 3) (10y + 3) We write the product of each factor that appears: (10y + 3)(10y 3)(100y 30y + 9) The highest power of each factor is one except of the factor of (10y 3); its highest power is Thus, L.C.D. (10y + 3)(10y 3) (100y 30y + 9) Concept #3 Writing Rational Expressions with the L.C.D. Now, let s put concepts #1 and # together. Find the L.C.D. of each pair of rational expressions and then write each one as an equivalent rational expression with the L.C.D. for the denominator: Ex. 9, 11 14a b 1ab 3 The L.C.D. of 14a b, 11 1ab 3 is 4a b 3.

5 Since 4a b 3 14a b 3b and 4a b 3 1ab 3 a, multiply the top and bottom of the first fraction by 3b and the top and bottom of the second by a: 3b 14a b 3b 11 a 1ab 3 a 1b 4a b 3 a 4a b 3 So, our fractions are 1b and a. 4a b 3 4a b 3 Ex. 10, 64x x 64x 3 1 (4x) 3 (1) 3 (4x 1)(16x + 4x + 1) 1 16x 16x + 1 1(16x 1) 1(4x 1)(4x + 1) The question becomes what to do with the negative one? But, the following are equivalent: 1 16x 1(4x 1)(4x+1) (4x 1)(4x+1) So, if we get a 1 in the denominator as part of the product, we can get rid of it by changing the sign of the numerator: So, our denominators are (4x 1)(16x + 4x + 1) & (4x 1)(4x + 1). We write the product of each factor that appears: (4x 1)(4x + 1)(16x + 4x + 1) The highest power of each factor is one. Thus, L.C.D. (4x 1)(4x + 1)(16x + 4x + 1) Since (4x 1)(4x + 1)(16x + 4x + 1) (4x 1)(16x + 4y + 1) (4x + 1) and (4x 1)(4x + 1)(16x + 4x + 1) (4x 1)(4x + 1) (16x + 4x + 1), multiply the top and bottom of the first fraction by (4x + 1) and the top and bottom of the second by (16x + 4x + 1): (4x+1) 64x 3 1 (4x 1)(16x +4x+1) (4x 1)(16x +4x+1) (4x+1) (4x+1) 8x+ (4x 1)(4x+1)(16x +4x+1) (4x 1)(4x+1)(16x +4x+1) 1 16x (4x 1)(4x+1) (4x 1)(4x+1) (16x +4x+ 1) (16x +4x+ 1) (16x +4x+ 1) (4x 1)(4x+1)(16x +4x+ 1) 80x 0x (4x 1)(4x+1)(16x +4x+ 1) 8

6 9 So, our fractions are 8x+ (4x 1)(4x+1)(16x +4x+1) & 80x 0x. (4x 1)(4x+1)(16x +4x+ 1) x Ex. 11, 6x +11x 3 8x +10x 63 6x + 11x 3 (3x )(x + ) 8x + 10x 63 (4x 9)(x + ) We write the product of each factor that appears: (3x )(x + )(4x 9) The highest power of each factor is one. Thus, L.C.D. (3x )(x + )(4x 9) Since (3x )(x + )(4x 9) (3x )(x + ) (4x 9) and Ex. 1 (3x )(x + )(4x 9) (4x 9)(x + ) (3x ), multiply the top and bottom of the first fraction by (4x 9) and the top and bottom of the second by (3x ): x 6x +11x 3 x (3x )(x+) x(4x 9) (3x )(x+)(4x 9) x (3x )(x+) (4x 9) (4x 9) 8x 18x (3x )(x+)(4x 9) 8x +10x 63 (4x 9)(x+) (4x 9)(x+) (3x ) (3x ) (3x ) (3x )(x+)(4x 9) So, our fractions are Since 9 x, 11 x 9 1x (3x )(x+)(4x 9) 8x 18x (3x )(x+)(4x 9) and 9 x x+9 (x 9) L.C.D. Our fractions are x 9 x 9 and 11 x 9. 1x (3x )(x+)(4x 9)., we do not need to find the

### Sect Definitions of a 0 and a n

5 Sect 5. - Definitions of a 0 and a n Concept # Definition of a 0. Let s examine the quotient rule when the powers are equal. Simplify: Ex. 5 5 There are two ways to view this problem. First, any non-zero

More information

### Section 2.4: Add and Subtract Rational Expressions

CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall

More information

### Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.

Chapter 7 Section 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational

More information

### At the end of this section, you should be able to solve equations that are convertible to equations in linear or quadratic forms:

Equations in Linear and Quadratic Forms At the end of this section, you should be able to solve equations that are convertible to equations in linear or quadratic forms: Equations involving rational expressions

More information

### Sect Introduction to Rational Expressions

127 Sect 7.1 - Introduction to Rational Expressions Concept #1 Definition of a Rational Expression. Recall that a rational number is any number that can be written as the ratio of two integers where the

More information

### Chapter 9 Notes SN AA U2C9

Chapter 9 Notes SN AA U2C9 Name Period Section 2-3: Direct Variation Section 9-1: Inverse Variation Two variables x and y show direct variation if y = kx for some nonzero constant k. Another kind of variation

More information

### Lesson 2. When the exponent is a positive integer, exponential notation is a concise way of writing the product of repeated factors.

Review of Exponential Notation: Lesson 2 - read to the power of, where is the base and is the exponent - if no exponent is denoted, it is understood to be a power of 1 - if no coefficient is denoted, it

More information

### Unit 3 Factors & Products

1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors

More information

### Chapter 5 Rational Expressions

Worksheet 4 (5.1 Chapter 5 Rational Expressions 5.1 Simplifying Rational Expressions Summary 1: Definitions and General Properties of Rational Numbers and Rational Expressions A rational number can be

More information

### Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

### All work must be shown or no credit will be awarded. Box all answers!! Order of Operations

Steps: All work must be shown or no credit will be awarded. Box all answers!! Order of Operations 1. Do operations that occur within grouping symbols. If there is more than one set of symbols, work from

More information

### Chapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010

Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression

More information

### Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

More information

### Assignment #1 MAT121 Summer 2015 NAME:

Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also

More information

### Chapter 4: Radicals and Complex Numbers

Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y

More information

### Sect Complex Numbers

161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a

More information

### x 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?

1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number

More information

### Exponents, Radicals, and Scientific Notation

General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

More information

### M098 Carson Elementary and Intermediate Algebra 3e Section 11.3

Objectives. Solve equations by writing them in quadratic form.. Solve equations that are quadratic in form by using substitution. Vocabulary Prior Knowledge Solve rational equations: Clear the fraction.

More information

### SYMBOL NAME DESCRIPTION EXAMPLES. called positive integers) negatives, and 0. represented as a b, where

EXERCISE A-1 Things to remember: 1. THE SET OF REAL NUMBERS SYMBOL NAME DESCRIPTION EXAMPLES N Natural numbers Counting numbers (also 1, 2, 3,... called positive integers) Z Integers Natural numbers, their

More information

### Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

### Answers to Sample Exam Problems

Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;

More information

### MATH 150 Pre-Calculus

MATH 150 Pre-Calculus Fall, 2014, WEEK 2 JoungDong Kim Week 2: 1D, 1E, 2A Chapter 1D. Rational Expression. Definition of a Rational Expression A rational expression is an expression of the form p, where

More information

### Chapter 3: Factors, Roots, and Powers

Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly

More information

### Partial Fraction Decomposition

Partial Fraction Decomposition As algebra students we have learned how to add and subtract fractions such as the one show below, but we probably have not been taught how to break the answer back apart

More information

### Unit 2 Boolean Algebra

Unit 2 Boolean Algebra 1. Developed by George Boole in 1847 2. Applied to the Design of Switching Circuit by Claude Shannon in 1939 Department of Communication Engineering, NCTU 1 2.1 Basic Operations

More information

### Simplifying Rational Expressions and Functions

Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written

More information

### Section 1.1 Notes. Real Numbers

Section 1.1 Notes Real Numbers 1 Types of Real Numbers The Natural Numbers 1,,, 4, 5, 6,... These are also sometimes called counting numbers. Denoted by the symbol N Integers..., 6, 5, 4,,, 1, 0, 1,,,

More information

### Section 8.3 Partial Fraction Decomposition

Section 8.6 Lecture Notes Page 1 of 10 Section 8.3 Partial Fraction Decomposition Partial fraction decomposition involves decomposing a rational function, or reversing the process of combining two or more

More information

### Real Numbers. Real numbers are divided into two types, rational numbers and irrational numbers

Real Numbers Real numbers are divided into two types, rational numbers and irrational numbers I. Rational Numbers: Any number that can be expressed as the quotient of two integers. (fraction). Any number

More information

### Reteach Multiplying and Dividing Rational Expressions

8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:

More information

### Chapter 7 Rational Expressions, Equations, and Functions

Chapter 7 Rational Expressions, Equations, and Functions Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions

More information

### Partial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254

Partial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254 Adding and Subtracting Rational Expressions Recall that we can use multiplication and common denominators to write a sum or difference

More information

### 5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality

5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality Now that we have studied the Addition Property of Equality and the Multiplication Property of Equality, we can solve

More information

### Axioms for the Real Number System

Axioms for the Real Number System Math 361 Fall 2003 Page 1 of 9 The Real Number System The real number system consists of four parts: 1. A set (R). We will call the elements of this set real numbers,

More information

### Reteach Variation Functions

8-1 Variation Functions The variable y varies directly as the variable if y k for some constant k. To solve direct variation problems: k is called the constant of variation. Use the known and y values

More information

### Ch. 12 Rational Functions

Ch. 12 Rational Functions 12.1 Finding the Domains of Rational F(n) & Reducing Rational Expressions Outline Review Rational Numbers { a / b a and b are integers, b 0} Multiplying a rational number by a

More information

«4 [< «

More information

### (4.2) Equivalence Relations. 151 Math Exercises. Malek Zein AL-Abidin. King Saud University College of Science Department of Mathematics

King Saud University College of Science Department of Mathematics 151 Math Exercises (4.2) Equivalence Relations Malek Zein AL-Abidin 1440 ه 2018 Equivalence Relations DEFINITION 1 A relation on a set

More information

### review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

More information

### MANY BILLS OF CONCERN TO PUBLIC

- 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -

More information

### Sect Exponents: Multiplying and Dividing Common Bases

154 Sect 5.1 - Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says

More information

### 8-5. A rational inequality is an inequality that contains one or more rational expressions. x x 6. 3 by using a graph and a table.

A rational inequality is an inequality that contains one or more rational expressions. x x 3 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = x and Y = 3. x The graph of Y1 is

More information

### REVIEW Chapter 1 The Real Number System

REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }

More information

### MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE. A rational expression is just a fraction involving polynomials, for example 3x2 2

MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE SECTION 1.2: PRECALCULUS REVIEW II Practice: 3, 7, 13, 17, 19, 23, 29, 33, 43, 45, 51, 57, 69, 81, 89 1. Rational Expressions and Other Algebraic Fractions A rational

More information

### 5.3. Polynomials and Polynomial Functions

5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a

More information

### LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

- 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

### Mission 1 Simplify and Multiply Rational Expressions

Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following

More information

### Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

More information

### Lesson #9 Simplifying Rational Expressions

Lesson #9 Simplifying Rational Epressions A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Factor the following epressions: A. 7 4 B. y C. y 49y Simplify: 5 5 = 4

More information

### Section 1.3 Review of Complex Numbers

1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that

More information

### MATH 110 SAMPLE QUESTIONS for Exam 1 on 9/9/05. 4x 2 (yz) x 4 y

MATH 110 SAMPLE QUESTIONS for Exam 1 on //05 1. (R. p7 #) After simplifying, the numerator of x x z yz 1 4 4x (yz) 1 3 x 4 y. (R.3 #11) The lengths of the legs of a right triangle are 7 and 4. Find the

More information

### 3.4. ZEROS OF POLYNOMIAL FUNCTIONS

3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find

More information

### LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

More information

### We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).

College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite

More information

### Review of Rational Expressions and Equations

Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis

More information

### Sect Addition and Subtraction of Polynomials

Sect 5.5 - Addition and Subtraction of Polynomials Concept #1 Introduction to Polynomials Before we begin discussing polynomials, let s review some items from chapter 1 with the following example: Complete

More information

### Chapter 2 : Boolean Algebra and Logic Gates

Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 1431-1432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean

More information

### LESSON 8.1 RATIONAL EXPRESSIONS I

LESSON 8. RATIONAL EXPRESSIONS I LESSON 8. RATIONAL EXPRESSIONS I 7 OVERVIEW Here is what you'll learn in this lesson: Multiplying and Dividing a. Determining when a rational expression is undefined Almost

More information

### LOWELL WEEKLY JOURNAL

Y G y G Y 87 y Y 8 Y - \$ X ; ; y y q 8 y \$8 \$ \$ \$ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 \$ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G

More information

### Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Epressions As a review, adding and subtracting fractions requires the fractions to have the same denominator. If they already have the same denominator, combine the numerators

More information

### AMC Lecture Notes. Justin Stevens. Cybermath Academy

AMC Lecture Notes Justin Stevens Cybermath Academy Contents 1 Systems of Equations 1 1 Systems of Equations 1.1 Substitution Example 1.1. Solve the system below for x and y: x y = 4, 2x + y = 29. Solution.

More information

### Chapter 7: Exponents

Chapter : Exponents Algebra Chapter Notes Name: Notes #: Sections.. Section.: Review Simplify; leave all answers in positive exponents:.) m -.) y -.) m 0.) -.) -.) - -.) (m ) 0.) 0 x y Evaluate if a =

More information

### Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic expressions.. Translate English phrases into algebraic expressions.. Determine whether a number is a solution

More information

### SECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are

SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers,

More information

### MA094 Part 2 - Beginning Algebra Summary

MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

More information

### Part 2 - Beginning Algebra Summary

Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian

More information

### Definition: A binary relation R from a set A to a set B is a subset R A B. Example:

Section 9.1 Rela%onships Relationships between elements of sets occur in many contexts. Every day we deal with relationships such as those between a business and its telephone number, an employee and his

More information

### MATHEMATICS 9 CHAPTER 7 MILLER HIGH SCHOOL MATHEMATICS DEPARTMENT NAME: DATE: BLOCK: TEACHER: Miller High School Mathematics Page 1

MATHEMATICS 9 CHAPTER 7 NAME: DATE: BLOCK: TEACHER: MILLER HIGH SCHOOL MATHEMATICS DEPARTMENT Miller High School Mathematics Page 1 Day 1: Creating expressions with algebra tiles 1. Determine the multiplication

More information

### Created by T. Madas MIXED SURD QUESTIONS. Created by T. Madas

MIXED SURD QUESTIONS Question 1 (**) Write each of the following expressions a single simplified surd a) 150 54 b) 21 7 C1A, 2 6, 3 7 Question 2 (**) Write each of the following surd expressions as simple

More information

### { independent variable some property or restriction about independent variable } where the vertical line is read such that.

Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with

More information

### Spring Nikos Apostolakis

Spring 07 Nikos Apostolakis Review of fractions Rational expressions are fractions with numerator and denominator polynomials. We need to remember how we work with fractions (a.k.a. rational numbers) before

More information

### Algebra 2 Honors: Final Exam Review

Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt

More information

### Ch. 12 Rational Functions

Ch. 12 Rational Functions 12.4 Simplifying Complex Expressions Outline Definition A fraction with a complex expression in the numerator and/or denominator. Methods for Simplifying Method 1 Division of

More information

### Math 154 :: Elementary Algebra

Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with

More information

### [Type the document subtitle] Math 0310

[Typethe document subtitle] Math 010 [Typethe document subtitle] Cartesian Coordinate System, Domain and Range, Function Notation, Lines, Linear Inequalities Notes-- Cartesian Coordinate System Page

More information

### CP Algebra 2 Unit 2-1: Factoring and Solving Quadratics WORKSHEET PACKET

CP Algebra Unit -1: Factoring and Solving Quadratics WORKSHEET PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor

More information

### YOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS

The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply

More information

### Part 1 - Pre-Algebra Summary Page 1 of 22 1/19/12

Part 1 - Pre-Algebra Summary Page 1 of 1/19/1 Table of Contents 1. Numbers... 1.1. NAMES FOR NUMBERS... 1.. PLACE VALUES... 3 1.3. INEQUALITIES... 4 1.4. ROUNDING... 4 1.5. DIVISIBILITY TESTS... 5 1.6.

More information

### Alg 1B Chapter 7 Final Exam Review

Name: Class: Date: ID: A Alg B Chapter 7 Final Exam Review Please answer all questions and show your work. Simplify ( 2) 4. 2. Simplify ( 4) 4. 3. Simplify 5 2. 4. Simplify 9x0 y 3 z 8. 5. Simplify 7w0

More information

### Section 2.8: The Power Chain Rule

calculus sin frontera Section 2.8: The Power Chain Rule I want to see what happens when I take a function I know, like g(x) = x 2, and raise it to a power, f (x) = ( x 2 ) a : f (x) = ( x 2 ) a = ( x 2

More information

### Algebraic Expressions

Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

### Unit 9 Study Sheet Rational Expressions and Types of Equations

Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by

More information

### REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate

More information

### Rational Expressions and Radicals

Rational Expressions and Radicals Rules of Exponents The rules for exponents are the same as what you saw in Section 5.1. Memorize these rules if you haven t already done so. x 0 1 if x 0 0 0 is indeterminant

More information

### SOLUTIONS FOR PROBLEMS 1-30

. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

### 1 h 9 e \$ s i n t h e o r y, a p p l i c a t i a n

T : 99 9 \ E \ : \ 4 7 8 \ \ \ \ - \ \ T \ \ \ : \ 99 9 T : 99-9 9 E : 4 7 8 / T V 9 \ E \ \ : 4 \ 7 8 / T \ V \ 9 T - w - - V w w - T w w \ T \ \ \ w \ w \ - \ w \ \ w \ \ \ T \ w \ w \ w \ w \ \ w \

More information

### 7.3 Adding and Subtracting Rational Expressions

7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add

More information

### THE RING OF POLYNOMIALS. Special Products and Factoring

THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products - rules

More information

### 0.Axioms for the Integers 1

0.Axioms for the Integers 1 Number theory is the study of the arithmetical properties of the integers. You have been doing arithmetic with integers since you were a young child, but these mathematical

More information

### A field trips costs \$800 for the charter bus plus \$10 per student for x students. The cost per student is represented by: 10x x

LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs \$800 for the charter bus plus \$10 per student

More information

### Tuesday, 3/28 : Ch. 9.8 Cubic Functions ~ Ch. 9 Packet p.67 #(1-6) Thursday, 3/30 : Ch. 9.8 Rational Expressions ~ Ch. 9 Packet p.

Ch. 9.8 Cubic Functions & Ch. 9.8 Rational Expressions Learning Intentions: Explore general patterns & characteristics of cubic functions. Learn formulas that model the areas of squares & the volumes of

More information

### Unit 2 Boolean Algebra

Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2

More information

### CONTENTS COLLEGE ALGEBRA: DR.YOU

1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.

More information

### CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions

Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,

More information

### Chapter 7: Exponents

Chapter : Exponents Algebra Chapter Notes Name: Algebra Homework: Chapter (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework M / Review of Sections.-.

More information

### Chapter 7: Exponents

Chapter 7: Exponents Algebra 1 Chapter 7 Notes Name: Algebra Homework: Chapter 7 (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework Section 7.:

More information

### Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it

Eby, MATH 010 Spring 017 Page 5 5.1 Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d)

More information

### Beginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College

Beginning Algebra Professor Sikora MAT002C POLYNOMIALS 6.1 Positive Integer Exponents x n = x x x x x [n of these x factors] base exponent Numerical: Ex: - = where as Ex: (-) = Ex: - = and Ex: (-) = Rule:

More information

### Fundamentals. Copyright Cengage Learning. All rights reserved.

Fundamentals Copyright Cengage Learning. All rights reserved. 1.2 Exponents and Radicals Copyright Cengage Learning. All rights reserved. Objectives Integer Exponents Rules for Working with Exponents Scientific

More information