5.1 Radical Notation and Rational Exponents

Size: px
Start display at page:

Download "5.1 Radical Notation and Rational Exponents"

Transcription

1 Section 5.1 Radical Notation and Rational Exponents Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots (such as square roots and cube roots). Along the way, we ll define higher roots and develop a few of their properties. Integer Exponents Recall that use of a positive integer exponent is simply a shorthand for repeated multiplication. For example, 2 3 = (5.1) In general, b n stands for the quanitity b multiplied by itself n times. With this definition, the following Laws of Exponents hold. Laws of Exponents 1. b r b s = b r+s 2. b r b s = br s 3. (b r ) s = b rs Recall that negative exponents, as well as the 0 exponent, are simply defined in such a way that the Laws of Exponents will work for all integer exponents. Negative exponents and the 0 exponent are defined as follows: Definition 1 provided that b 0. b n = 1 b n and b 0 = 1 For example, 2 0 = 1, and 2 4 = 1/2 4 = 1/16. We now have b n defined for all integers n, in such a way that the Laws of Exponents hold. Recall that we can likewise define expressions using rational exponents, such as 2 1/3, in a consistent manner. Before doing reviewing that material, we ll need to take a detour and review roots. Roots Square Roots: Let s begin by reminding ourselves what the square root of a real number represents.

2 2 Chapter 5 Definition 2 Given a real number a, the square root of a is a number x such that x 2 = a. We investigate square roots in more detail by looking for solutions of the equation x 2 = a. (5.2) There are three cases, each depending on the value and sign of a. Summary: Square Roots The solutions of x 2 = a are called square roots of a. Case I: a < 0. The equation x 2 = a has no real solutions. Case II: a = 0. The equation x 2 = a has one real solution, namely x = 0. Thus, 0 = 0. Case III: a > 0. The equation x 2 = a has two real solutions, x = ± a. The notation a calls for the positive square root of a, that is, the positive solution of x 2 = a. The notation a calls for the negative square root of a, that is, the negative solution of x 2 = a. Let s look at some examples. Example 1 What are the solutions of x 2 = 5? We re asked to find a solution of x 2 = 5, so you must find a number whose square equals 5. However, whenever you square a real number, the result is always nonnegative (zero or positive). It is not possible to square a real number and get 5. That is, 5 is not a real number. Example 2 What are the solutions of x 2 = 0? There is only one solution, namely x = 0. Note that this means that 0 = 0. Example 3 What are the solutions of x 2 = 25?

3 Section 5.1 Radical Notation and Rational Exponents 3 The solutions of x 2 = 25 are called square roots of 25 and are written x = ± 25. In this case, we can simplify further and write x = ±5. Cube Roots: Let s move on to the definition of cube roots. Definition 3 Given a real number a, a cube root of a is a number x such that x 3 = a. For example, 2 is a cube root of 8 since 2 3 = 8. Likewise, 4 is a cube root of 64 since ( 4) 3 = 64. Thus, taking the cube root is the opposite of cubing, so the definition of cube root must be closely connected to the graph of y = x 3, the cubing function. Therefore, we look for solutions of A detailed summary of cube roots follows. x 3 = a. (5.3) Summary: Cube Roots The solutions of x 3 = a are called the cube roots of a. Whether a is negative, zero, or positive makes no difference. There is exactly one real solution, namely x = 3 a. Let s look at an example. Example 4 What are the solutions of x 3 = 8? The equation x 3 = 8 has exactly one real solution, denoted x = 3 8. Now since ( 2) 3 = 8, it follows that x = 2 is a real solution of x 3 = 8. Consequently, the cube root of 8 is 2, and we write 3 8 = 2. Again, because there is only one real solution of x 3 = 8, the notation 3 8 is pronounced the cube root of 8. Note that, unlike the square root of a negative number, the cube root of a negative number is allowed.

4 4 Chapter 5 Higher Roots: The previous discussions generalize easily to higher roots, such as fourth roots, fifth roots, sixth roots, etc. The key is to remember that all even roots behave the same as the square root and all odd roots behave the same as the cube root. Summary: Even nth Roots If n is a positive even integer, then the solutions of x n = a are called nth roots of a. Case I: a < 0. The equation x n = a has no real solutions. Case II: a = 0. The equation x n = a has exactly one real solution, namely x = 0. Thus, n 0 = 0. Case III: a > 0. The equation x n = a has two real solutions, x = ± n a. The notation n a calls for the positive nth root of a, that is, the positive solution of x n = a. The notation n a calls for the negative nth root of a, that is, the negative solution of x n = a. Summary: Odd nth Roots If n is a positive odd integer, then the solutions of x n = a are called the nth roots of a. Whether a is negative, zero, or positive makes no difference. There is exactly one real solution of x n = a, denoted x = n a. Rational Exponents Recall that rational exponents are defined in such a way that the Laws of Exponents still apply. For example, if n is any odd positive integer, then 2 1/n must be defined by the formula 2 1 n = n 2. With this definition, the Laws of Exponents hold for all rational exponents. Definition 4 For a positive rational exponent m n, and b > 0, For a negative rational exponent m n, b m n = n b m = ( n b) m. (5.4) b m 1 n =. (5.5) b m n For b < 0, the same definitions make sense only when n is odd. For example ( 2) 1 4 is not defined.

5 Section 5.1 Radical Notation and Rational Exponents 5 Remark 1. example ( 2) 1 4 Example 5 For b < 0, the same definitions make sense only when n is odd. For is not defined. Simplify the following expressions, and write them in the form x r : a) x x 4 = x = x = x 12 a) x 2 3 x 1 4, b) x 2 3 x 1 4, c) ( ) x b) x 2 3 x 1 4 c) ( x 2 3 = x = x = x 5 12 ) 1 4 = x = x 2 12 = x 1 6 Multiplication Properties of Radicals Recall the following property of radicals. Property 1 Let a and b be positive real numbers. Then, ab = a b. (5.6) This result can be used in two distinctly different ways. You can use the result to multiply two square roots, as in 7 5 = 35. You can also use the result to factor, as in 35 = 5 7. One question that arises is when is a radical in reduced form? We use the following set of guidelines for a special form of the answer which we will call simple radical form. The First Guideline for Simple Radical Form. When possible, factor out a perfect square. For example, 32 is not in simple radical form, as it is possible to factor out a perfect square, as in 32 = 16 2 = 4 2.

6 6 Chapter 5 Let s place another radical expression in simple radical form. Example 6 Place 50 in simple radical form. Because 50 = 25 2, we can use Property 1 to write 50 = 25 2 = 5 2. When working with square roots it can be extremely useful to recall the definition of a rational exponent. The process of placing a square root into simple radical form can be done much more efficiently using rational exponents. Taking the Square Root of an Even Power. When taking a square root of x n, when x is a positive real number and n is an even natural number, divide the exponent by two. In symbols, x n = x n/2. Also, recall that raising a product to a power requires that we raise each factor to that power. Raising a Product to a Power. (ab) n = a n b n. Let s look at an example that employs these properties. Example 7 Simplify In this example, the difficulty is the fact that the exponents are not divisible by 2. However, if possible, the first guideline of simple radical form requires that we factor out a perfect square. So, extract each factor raised to the highest possible power that is divisible by 2, as in = Now, divide each exponent by = Finally, simplify by expanding each exponential factor and multiplying = = 12 6

7 Section 5.1 Radical Notation and Rational Exponents 7 When there are variables in the radicand, then we need to be careful how we simplify. Note that if we incorrectly reduce ( 3) 2, we might be tempted to say it is 3. But recall that ( 3) 2 is a positive square root and the resulting simplification also needs to be positive. This discussion leads to the following result. The Positive Square Root of the Square of x. If x is any real number, then x 2 = x. Let s use these ideas to simplify some radical expressions that contain variables. Variable Expressions Example 8 Given that the x represents any real numbers, place the radical expression 48x 6 in simple radical form. Simple radical form demands that we factor out a perfect square, if possible. In this case, 48 = 16 3 and we factor out the highest power of x that is divisible by 2. 48x 6 = 16x 6 3 We can now use Property 1 to take the square root of each factor. 16x 6 3 = 16 x 6 3 Now, remember that the notation calls for a nonnegative square root, so we must insure that each response in the equation above is nonnegative. Thus, 16 x 6 3 = 4 x 3 3. Division Properties of Radicals We continue our review by stating the following property of radicals. Property 2 Let a and b be positive real numbers. Then, a a b =. b

8 8 Chapter 5 Simple Radical Form Continued With this property we can add two more guidelines for simple radical form. Simple Radical Form. When your answer is a radical expression: 1. If possible, factor out a perfect square. 2. Don t leave fractions under a radical. 3. Don t leave radicals in the denominator of a fraction. If x is any real number, recall again that x 2 = x. Let s look at another example. Example 9 Place the expression 18/x 6 in simple radical form. Discuss the domain. Note that x cannot equal zero, otherwise the denominator of 18/x 6 would be zero, which is not allowed. However, whether x is positive or negative, x 6 will be a positive number (raising a nonzero number to an even power always produces a positive real number), and 18/x 6 is well-defined. Keeping in mind that x is nonzero, but could either be positive or negative, we proceed by first invoking Property 2, taking the positive square root of both numerator and denominator of our radical expression x 6 = x 6 From the numerator, we factor a perfect square. In the denominator, we use absolute value bars to insure a positive square root. 18 x 6 = 9 2 x 3 = 3 2 x 3 We can use the Product Rule for Absolute Value to write x 3 = x 2 x = x 2 x. Note that we do not need to wrap x 2 in absolute value bars because x 2 is already positive. 3 2 x 3 = 3 2 x 2 x Because x could be positive or negative, we cannot remove the absolute value bars around x. We are done.

9 Section 5.1 Radical Notation and Rational Exponents 9 Radical Expressions Now that we have reviewed how to multiply and divide square roots, we will simplify a number of more extensive expressions containing square roots, Example 10 Simplify the expression (2 12)(3 3). Place your answer in simple radical form. We ll first take the product of 2 and 3, then the product of 12 and 3, then multiply these results together. (2 12)(3 3) = (2 3)( 12 3) = 6 36 Of course, 36 = 6, so we can simplify further = 6 6 = 36 Recall the following operation, where you distribute the 2, multiplying each term in the parentheses by 2. 2(3 + x) = 6 + 2x You can do precisely the same thing with radical expressions. 2(3 + 5) = Like the familiar example above, we distributed the 2, multiplying each term in the parentheses by 2. Let s look at an example. Example 11 Use the distributive property to expand the expression 12(3 + 3), placing your final answer in simple radical form. First, distribute the 12, multiplying each term in the parentheses by 12. Note that 12 3 = (3 + 3) = = However, this last expression is not in simple radical form, as we can factor out a perfect square (12 = 4 3) = 3( 4 3) + 6 = 3(2 3) + 6 =

10 10 Chapter 5 The distributive property is also responsible in helping us combine like terms. For example we know that 3x + 5x = 8x. It is the distributive property that actually provides this solution. Note how we use the distributive property to factor x from each term. 3x + 5x = (3 + 5)x Hence, 3x + 5x = 8x. You can do the same thing with radical expressions = (3 + 5) 2 Hence, = 8 2, and the structure of this result is identical to that shown in 3x + 5x = 8x. There is no difference in the way we combine these like terms. We repeat the common factor and add coefficients. In the case that we don t have like terms, as in 3x + 5y, there is nothing to be done. In like manner, each of the following expressions have no like terms that you can combine. They are as simplified as they are going to get , , and 2 x + 5 y However, there are times when it can look as if you don t have like terms, but when you place everything in simple radical form, you discover that you do have like terms that can be combined by adding coefficients. Example 12 Simplify the expression , placing the final expression in simple radical form. We can extract a perfect square (27 = 9 3) = 5( 9 3) = 5(3 3) = Note that we now have like terms that can be combined by adding coefficients = 23 3

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

1.6 The Order of Operations

1.6 The Order of Operations 1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

More information

Exponents and Radicals

Exponents and Radicals Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

More information

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

More information

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

More information

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (-8) (-8)(-8) 64 Make sure students realize that x means (x ), not (-x).

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

Radicals - Multiply and Divide Radicals

Radicals - Multiply and Divide Radicals 8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

Simplification of Radical Expressions

Simplification of Radical Expressions 8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of

More information

8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz

8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz 8-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS 23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

Radicals - Rationalize Denominators

Radicals - Rationalize Denominators 8. Radicals - Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers. 1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Irrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.

Irrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that

More information

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes) NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

More information

Exponents, Radicals, and Scientific Notation

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

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

SIMPLIFYING SQUARE ROOTS EXAMPLES

SIMPLIFYING SQUARE ROOTS EXAMPLES SIMPLIFYING SQUARE ROOTS EXAMPLES 1. Definition of a simplified form for a square root The square root of a positive integer is in simplest form if the radicand has no perfect square factor other than

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

More information

Unit 7: Radical Functions & Rational Exponents

Unit 7: Radical Functions & Rational Exponents Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

More information

Square Roots and Other Radicals

Square Roots and Other Radicals Radicals - Definition Radicals, or roots, are the opposite operation of applying exponents. A power can be undone with a radical and a radical can be undone with a power. For example, if you square 2,

More information

Negative Integer Exponents

Negative Integer Exponents 7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

More information

Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Radicals - Rational Exponents

Radicals - Rational Exponents 8. Radicals - Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013 Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

More information

SIMPLIFYING SQUARE ROOTS

SIMPLIFYING SQUARE ROOTS 40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Math 115 Spring 2011 Written Homework 5 Solutions

Math 115 Spring 2011 Written Homework 5 Solutions . Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers: Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Common Core Standards for Fantasy Sports Worksheets. Page 1

Common Core Standards for Fantasy Sports Worksheets. Page 1 Scoring Systems Concept(s) Integers adding and subtracting integers; multiplying integers Fractions adding and subtracting fractions; multiplying fractions with whole numbers Decimals adding and subtracting

More information

Georgia Standards of Excellence Curriculum Frameworks. Mathematics. GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited

Georgia Standards of Excellence Curriculum Frameworks. Mathematics. GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited These materials are for nonprofit educational purposes only. Any other use

More information

3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

More information

MATH-0910 Review Concepts (Haugen)

MATH-0910 Review Concepts (Haugen) Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

More information

2.3 Solving Equations Containing Fractions and Decimals

2.3 Solving Equations Containing Fractions and Decimals 2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Simplifying Square-Root Radicals Containing Perfect Square Factors

Simplifying Square-Root Radicals Containing Perfect Square Factors DETAILED SOLUTIONS AND CONCEPTS - OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Radicals - Square Roots

Radicals - Square Roots 8.1 Radicals - Square Roots Objective: Simplify expressions with square roots. Square roots are the most common type of radical used. A square root unsquares a number. For example, because 5 2 = 25 we

More information

Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková

Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

More information

2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations 2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Powers and Roots. 20 Sail area 810 ft 2. Sail area-displacement ratio (r) 22 24 26 28 30 Displacement (thousands of pounds)

Powers and Roots. 20 Sail area 810 ft 2. Sail area-displacement ratio (r) 22 24 26 28 30 Displacement (thousands of pounds) C H A P T E R Powers and Roots Sail area-displacement ratio (r) 1 16 14 1 1 Sail area 1 ft 4 6 Displacement (thousands of pounds) ailing the very word conjures up images of warm summer S breezes, sparkling

More information

Pre-Algebra - Order of Operations

Pre-Algebra - Order of Operations 0.3 Pre-Algebra - Order of Operations Objective: Evaluate expressions using the order of operations, including the use of absolute value. When simplifying expressions it is important that we simplify them

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0. Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

More information

Tool 1. Greatest Common Factor (GCF)

Tool 1. Greatest Common Factor (GCF) Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

More information

CAHSEE on Target UC Davis, School and University Partnerships

CAHSEE on Target UC Davis, School and University Partnerships UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

More information

Accentuate the Negative: Homework Examples from ACE

Accentuate the Negative: Homework Examples from ACE Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),

More information

Lesson 9: Radicals and Conjugates

Lesson 9: Radicals and Conjugates Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

More information

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

Chapter 4 -- Decimals

Chapter 4 -- Decimals Chapter 4 -- Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789

More information

7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2

COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2 COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what

More information

Order of Operations More Essential Practice

Order of Operations More Essential Practice Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

Section 1.1 Real Numbers

Section 1.1 Real Numbers . Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order 26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

More information

Lesson 9: Radicals and Conjugates

Lesson 9: Radicals and Conjugates Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

More information

Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.

Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. 31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The

More information

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

More information

Domain of a Composition

Domain of a Composition Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

More information

Linear Equations and Inequalities

Linear Equations and Inequalities Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

is the degree of the polynomial and is the leading coefficient.

is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

More information

More Quadratic Equations

More Quadratic Equations More Quadratic Equations Math 99 N1 Chapter 8 1 Quadratic Equations We won t discuss quadratic inequalities. Quadratic equations are equations where the unknown appears raised to second power, and, possibly

More information

QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write

More information

Multiplying and Dividing Radicals

Multiplying and Dividing Radicals 9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated

More information

Solving Rational Equations and Inequalities

Solving Rational Equations and Inequalities 8-5 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations

More information

Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

More information