Not For Sale. A Review of Basic Algebra. CAREERS AND MATHEMATICS: Pharmacist

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1 A Review of Basic Algebra A Istockphoto.com/Lucas Rucchin CAREERS AND MATHEMATICS: Pharmacist Pharmacists distribute prescription drugs to individuals. They also advise patients, physicians, and other healthcare workers on the selection, dosages, interactions, and side effects of medications. They also monitor patients to ensure that they are using their medications safely and effectively. Some pharmacists specialize in oncology, nuclear pharmacy, geriatric pharmacy, or psychiatric pharmacy. Education and Mathematics Required Pharmacists are required to possess a Pharm.D. degree from an accredited college or school of pharmacy. This degree generally takes four years to complete. To be admitted to a Pharm.D. program, at least two years of college must be completed, which includes courses in the natural sciences, mathematics, humanities, and the social sciences. A series of eaminations must also be passed to obtain a license to practice pharmacy. College Algebra, Trigonometry, Statistics, and Calculus I are courses required for admission to a Pharm.D. program. How Pharmacists Use Math and Who Employs Them Pharmacists use math throughout their work to calculate dosages of various drugs. These dosages are based on weight and whether the medication is given in pill form, by infusion, or intravenously. Most pharmacists work in a community setting, such as a retail drugstore, or in a healthcare facility, such as a hospital. Career Outlook and Salary Employment of pharmacists is epected to grow by 7 percent between 2008 and 208, which is faster than the average for all occupations. Median annual wages of wage and salary pharmacists is approimately $06,40. For more information see: A. Sets of Real Numbers A.2 Integer Eponents and Notation A.3 Rational Eponents and Radicals A.4 Polynomials A. Factoring Polynomials A.6 Rational Epressions Appendi Review Appendi Test In this appendi, we review many concepts and skills learned in previous algebra courses. Not For Sale A

2 Not For Sale A2 Appendi A A Review of Basic Algebra A. Sets of Real Numbers In this section, we will learn to. Identify sets of real numbers. 2. Identify properties of real numbers. 3. Graph subsets of real numbers on the number line. 4. Graph intervals on the number line.. Define absolute value. 6. Find distances on the number line SUDOKU example Sudoku, a game that involves number placement, is very popular. The objective is to fill a 9 by 9 grid so that each column, each row and each of the 3 by 3 blocks contains the numbers from to 9. A partially completed Sudoku grid is shown in the margin. To solve Sudoku puzzles, logic and the set of numbers,, 2, 3, 4,, 6, 7, 8, 96 are used. Sets of numbers are important in mathematics, and we begin our study of algebra with this topic. A set is a collection of objects, such as a set of dishes or a set of golf clubs. The set of vowels in the English language can be denoted as a, e, i, o, u6, where the braces 6 are read as the set of. If every member of one set B is also a member of another set A, we say that B is a subset of A. We can denote this by writing B ( A, where the symbol ( is read as is a subset of. (See Figure A. below.) If set B equals set A, we can write B 8 A. If A and B are two sets, we can form a new set consisting of all members that are in set A or set B or both. This set is called the union of A and B. We can denote this set by writing, A c B where the symbol c is read as union. (See Figure A. below.) We can also form the set consisting of all members that are in both set A and set B. This set is called the intersection of A and B. We can denote this set by writing A d B, where the symbol d is read as intersection. (See Figure A. below.) B B A A A A B B FIGURE A- A A B Understanding Subsets and Finding the Union and Intersection of Two Sets Let A a, e, i6, B c, d, e6, and V a, e, i, o, u6. a. Is A ( V? b. Find A c B. c. Find A d B. solution a. Since each member of set A is also a member of set V, A ( V. b. The union of set A and set B contains the members of set A, set B, or both. Thus, A c B a, c, d, e, i6. c. The intersection of set A and set B contains the members that are in both set A and set B. Thus, A d B e6. B self Check a. Is B ( V? b. Find B c V c. Find A d V Now Try Eercise 33.

3 Section A. Sets of Real Numbers A3. Identify Sets of Real Numbers There are several sets of numbers that we use in everyday life. Caution Basic Sets of Numbers Remember that the denominator of a fraction can never be 0. Natural numbers The numbers that we use for counting:, 2, 3, 4,, 6, % 6 Whole numbers The set of natural numbers including 0: 0,, 2, 3, 4,, 6, % 6 Integers The set of whole numbers and their negatives: %, 2, 24, 23, 22, 2, 0,, 2, 3, 4,, % 6 In the definitions above, each group of three dots (called an ellipsis) indicates that the numbers continue forever in the indicated direction. Two important subsets of the natural numbers are the prime and composite numbers. A prime number is a natural number greater than that is divisible only by itself and. A composite number is a natural number greater than that is not prime. The set of prime numbers: 2, 3,, 7,, 3, 7, 9, 23, 29, 3, % 6 The set of composite numbers: 4, 6, 8, 9, 0, 2, 4,, 6, 8, 20, 2, % 6 Two important subsets of the set of integers are the even and odd integers. The even integers are the integers that are eactly divisible by 2. The odd integers are the integers that are not eactly divisible by 2. The set of even integers: %, 20, 28, 26, 24, 22, 0, 2, 4, 6, 8, 0, % 6 The set of odd integers: %, 29, 27, 2, 23, 2,, 3,, 7, 9, % 6 So far, we have listed numbers inside braces to specify sets. This method is called the roster method. When we give a rule to determine which numbers are in a set, we are using set-builder notation. To use set-builder notation to denote the set of prime numbers, we write 0 is a prime number6 variable such that rule that determines membership in the set Read as the set of all numbers such that is a prime number. Recall that when a letter stands for a number, it is called a variable. The fractions of arithmetic are called rational numbers. rational Numbers Rational numbers are fractions that have an integer numerator and a nonzero integer denominator. Using set-builder notation, the rational numbers are e a 0 a is an integer and b is a nonzero integer f b Rational numbers can be written as fractions or decimals. Some eamples of rational numbers are, , %, % The = sign indicates Not For Sale that two quantities are equal.

4 Not For Sale A4 Appendi A A Review of Basic Algebra These eamples suggest that the decimal forms of all rational numbers are either terminating decimals or repeating decimals. EXAMPLE 2 solution self Check 2 real Numbers determining whether the Decimal Form of a Fraction Terminates or repeats Determine whether the decimal form of each fraction terminates or repeats: a. 7 6 b In each case, we perform a long division and write the quotient as a decimal. a. To change to a decimal, we perform a long division to get Since terminates, we can write 7 6 as a terminating decimal. b. To change to a decimal, we perform a long division to get %. Since repeats, we can write 6 99 as a repeating decimal. We can write repeating decimals in compact form by using an overbar. For eample, % 0.6. Determine whether the decimal form of each fraction terminates or repeats: a. 38 b Now Try Eercise 3. Some numbers have decimal forms that neither terminate nor repeat. These nonterminating, nonrepeating decimals are called irrational numbers. Three eamples of irrational numbers are % " % and p % The union of the set of rational numbers (the terminating and repeating decimals) and the set of irrational numbers (the nonterminating, nonrepeating decimals) is the set of real numbers (the set of all decimals). A real number is any number that is rational or irrational. Using set-builder notation, the set of real numbers is 0 is a rational or an irrational number6 EXAMPLE 3 solution Classifying Real Numbers In the set 23, 22, 0, 2,, ", 2, 4,, 66, list all a. even integers b. prime numbers c. rational numbers We will check to see whether each number is a member of the set of even integers, the set of prime numbers, and the set of rational numbers. a. even integers: 2, 0, 2, 4, 6 b. prime numbers: 2, c. rational numbers: 3, 2, 0, 2,, 2, 4,, 6

5 Section A. Sets of Real Numbers A self Check 3 In the set in Eample 3, list all a. odd integers b. composite numbers c. irrational numbers. Now Try Eercise 43. Figure A-2 shows how the previous sets of numbers are related. Real numbers 8 3,, 2, 0,, π, Properties of Real Numbers Noninteger rational numbers 3, 2 0.,,, Rational numbers 6,.2, 0, 2, 3, Negative integers 47, 7,, Integers 4,, 0, 2 Zero 0 FIGURE A-2 2. Identify Properties of Real Numbers Whole numbers 0,, 4, 8, 0, 3, 0 Irrational numbers, π, 2, 2 0 Natural numbers (positive integers), 2, 38, 990 When we work with real numbers, we will use the following properties. If a, b, and c are real numbers, The Commutative Properties for Addition and Multiplication a b b a ab ba The Associative Properties for Addition and Multiplication a b2 c a b c2 ab2c abc2 The Distributive Property of Multiplication over Addition or Subtraction ab c2 ab ac or ab 2 c2 ab 2 ac The Double Negative Rule 22a2 a Caution When the Associative Property is used, the order of the real numbers does not change. The real numbers that occur within the parentheses change. Not The For Distributive Property Sale also applies when more than two terms are within parentheses.

6 Not For Sale A6 Appendi A A Review of Basic Algebra EXAMPLE 4 Identifying Properties of Real Numbers SOLUTION a b. 3 y y 3? 2 We will compare the form of each statement to the forms listed in the properties of real numbers bo. a. This form matches the Associative Property of Addition. b. This form matches the Distributive Property. Self Check 4 Comment Zero is neither positive nor negative. Comment "2 can be shown as the diagonal of a square with sides of length. a. mn nm b. y2z yz2 c. p q q p Now Try Eercise Graph Subsets of Real Numbers on the Number Line We can graph subsets of real numbers on the number line. The number line shown in Figure A-3 continues forever in both directions. The positive numbers are represented by the points to the right of 0, and the negative numbers are represented by the points to the left of 0. Negative numbers Positive numbers FIGURE A-3 Figure A-4(a) shows the graph of the natural numbers from to. The point associated with each number is called the graph of the number, and the number is called the coordinate of its point. Figure A-4(b) shows the graph of the prime numbers that are less than 0. Figure A-4(c) shows the graph of the integers from 24 to 3. Figure A-4(d) shows the graph of the real numbers 2 7 3, 23 4, 0.3, and " (a) (b) 7/3 3/ (c) FIGURE A-4 (d) The graphs in Figure A-4 suggest that there is a one-to-one correspondence between the set of real numbers and the points on a number line. This means that to each real number there corresponds eactly one point on the number line, and to each point on the number line there corresponds eactly one real-number coordinate. EXAMPLE Graphing a Set of Numbers on a Number Line Graph the set U23, 2 4 3, 0, "V.

7 Section A. Sets of Real Numbers A7 solution We will mark (plot) each number on the number line. To the nearest tenth, " / self Check Graph the set U22, 3 4, "3V. (Hint: To the nearest tenth, "3.7.) Now Try Eercise. 4. Graph Intervals on the Number Line To show that two quantities are not equal, we can use an inequality symbol. Symbol Read as Eamples 2 is not equal to 2 8 and , is less than 2, 20 and 0.7,.. is greater than. 9 and # is less than or equal to 2 # 2 and.7 # 2.3 $ is greater than or equal to 9 $ 9 and.2 $ 3.7 < is approimately equal to "2 <.44 and "3 <.732 It is possible to write an inequality with the inequality symbol pointing in the opposite direction. For eample, 2, 20 is equivalent to $ 2.7 is equivalent to 2.7 # 2.3 In Figure A-3, the coordinates of points get larger as we move from left to right on a number line. Thus, if a and b are the coordinates of two points, the one to the right is the greater. This suggests the following facts: If a. b, point a lies to the right of point b on a number line. If a, b, point a lies to the left of point b on a number line. Figure A-(a) shows the graph of the inequality. 22 (or 22, ). This graph includes all real numbers that are greater than 22. The parenthesis at 22 indicates that 22 is not included in the graph. Figure A-(b) shows the graph of # (or $ ). The bracket at indicates that is included in the graph. ( 2 ] (a) FIGURE A- (b) Sometimes two inequalities can be written as a single epression called a compound inequality. For eample, the compound inequality,, 2 is a combination of the inequalities, and, 2. It is read as is less than, and is less than 2, and it means that is between and 2. Its graph is shown in Figure A-6. ( ) Not For Sale 2 FIGURE A-6

8 Not For Sale A8 Appendi A A Review of Basic Algebra The graphs shown in Figures A- and A-6 are portions of a number line called intervals. The interval shown in Figure A-7(a) is denoted by the inequality 22,, 4, or in interval notation as 22, 42. The parentheses indicate that the endpoints are not included. The interval shown in Figure A-7(b) is denoted by the inequality., or as, `2 in interval notation. ( ) 2 4 ( Caution EXAMPLE 6 ( 2, 4) 2 < < 4 is between 2 and 4. (a) FIGURE A-7 (, ) > is greater than. The symbol ` (infinity) is not a real number. It is used to indicate that the graph in Figure A-7(b) etends infinitely far to the right. A compound inequality such as 22,, 4 can be written as two separate inequalities:. 22 and, 4 This epression represents the intersection of two intervals. In interval notation, this epression can be written as 22, `2 d 2`, 42 Read the symbol d as intersection. Since the graph of 22,, 4 will include all points whose coordinates satisfy both. 22 and, 4 at the same time, its graph will include all points that are larger than 22 but less than 4. This is the interval 22, 42, whose graph is shown in Figure A-7(a). Writing an Inequality in Interval Notation and Graphing the Inequality Write the inequality 23,, in interval notation and graph it. solution This is the interval 23, 2. Its graph includes all real numbers between 23 and, as shown in Figure A-8. (b) ( ) 3 FIGURE A-8 Self Check 6 Write the inequality 22, # in interval notation and graph it. Now Try Eercise 63. If an interval etends forever in one direction, it is called an unbounded interval.

9 Section A. Sets of Real Numbers A9 ( [ ) 2 3 [ 2, 3) 2 < 3 (b) ) 3 2 ( 3, 2) 3 < < 2 (a) FIGURE A-9 [ ] 2 4 [ 2, 4] 2 4 FIGURE A-0 Unbounded Intervals Interval Inequality Graph a, `2. a ( a 2, `2. 2 ( a, `2 $ a [ a 32, `2 $ 2 [ `, a2, a a 2`, 22, 2 ) `, a4 # a 2`, 24 # 2 ] `, `2 2`,, ` A bounded interval with no endpoints is called an open interval. Figure A-9(a) shows the open interval between 23 and 2. A bounded interval with one endpoint is called a half-open interval. Figure A-9(b) shows the half-open interval between 22 and 3, including 22. Intervals that include two endpoints are called closed intervals. Figure A-0 shows the graph of a closed interval from 22 to 4. Open Intervals, Half-Open Intervals, and Closed Intervals Interval Inequality Graph Open a, b2 a,, b ( ) a b 22, 32 22,, 3 ( ) Half-Open 3a, b2 a #, b [ ) a b 322, #, 3 [ ) Half-Open a, b4 a, # b ( ] a b 22, 34 22, # 3 ( ] Closed 3a, b4 a # # b [ ] a b 322, # # 3 [ ] a ( [ Not For Sale

10 Not For Sale A0 Appendi A A Review of Basic Algebra EXAMPLE 7 solution Writing an Inequality in Interval Notation and Graphing the Inequality Write the inequality 3 # in interval notation and graph it. The inequality 3 # can be written in the form $ 3. This is the interval 33, `2. Its graph includes all real numbers greater than or equal to 3, as shown in Figure A-. [ 3 FIGURE A- self Check 7 EXAMPLE 8 solution self Check 8 Write the inequality. in interval notation and graph it. now Try Eercise 6. Writing an Inequality in Interval Notation and Graphing the Inequality Write the inequality $ $ 2 in interval notation and graph it. The inequality $ $ 2 can be written in the form 2 # # This is the interval 32, 4. Its graph includes all real numbers from 2 to. The graph is shown in Figure A-2. [ ] FIGURE A-2 Write the inequality 0 # # 3 in interval notation and graph it. Now Try Eercise 69. The epression, 22 or $ 3 Read as is less than 22 or is greater than or equal to 3. represents the union of two intervals. In interval notation, it is written as 2`, 222 c 33, `2 Read the symbol c as union. Its graph is shown in Figure A-3. ) 2 FIGURE A-3 [ 3. Define Absolute Value The absolute value of a real number (denoted as 0 0 ) is the distance on a number line between 0 and the point with a coordinate of. For eample, points with coordinates of 4 and 24 both lie four units from 0, as shown in Figure A-4. Therefore, it follows that units 4 units FIGURE A-4

11 Section A. Sets of Real Numbers A In general, for any real number, We can define absolute value algebraically as follows. absolute Value If is a real number, then 0 0 when $ when, 0 Caution Remember that is not always positive and 2 is not always negative. EXAMPLE 9 solution self Check 9 EXAMPLE 0 solution This definition indicates that when is positive or 0, then is its own absolute value. However, when is negative, then 2 (which is positive) is its absolute value. Thus, 0 0 is always nonnegative. 0 0 $ 0 for all real numbers Using the Definition of Absolute Value Write each number without using absolute value symbols: a b c d In each case, we will use the definition of absolute value. a b c d Write each number without using absolute value symbols: a b c Now Try Eercise 8. In Eample 0, we must determine whether the number inside the absolute value is positive or negative. Simplifying an Epression with Absolute Value Symbols Write each number without using absolute value symbols: a. 0 p 2 0 b p 0 c if $ a. Since p < 3.46, p 2 is positive, and p 2 is its own absolute value. 0 p 2 0 p 2 b. Since 2 2 p is negative, its absolute value is 22 2 p p p p2 22 p p 2 2 c. Since $, the epression 2 2 is negative, and its absolute value is provided $ self Check 0 Write each number without using absolute value symbols. Hint: " < a. P 2 2 " P b if # Not For Sale Now Try Eercise 89.

12 Not For Sale A2 Appendi A A Review of Basic Algebra 6. Find Distances on the Number Line On the number line shown in Figure A-, the distance between the points with coordinates of and 4 is 4 2, or 3 units. However, if the subtraction were done in the other order, the result would be 2 4, or 23 units. To guarantee that the distance between two points is always positive, we can use absolute value symbols. Thus, the distance d between two points with coordinates of and 4 is d d = 4 = 3 distance between Two Points EXAMPLE solution FIGURE A- In general, we have the following definition for the distance between two points on the number line. If a and b are the coordinates of two points on the number line, the distance between the points is given by the formula d 0 b 2 a 0 Finding the Distance between Two Points on a Number Line Find the distance on a number line between points with coordinates of a. 3 and b. 22 and 3 c. 2 and 2 We will use the formula for finding the distance between two points. a. d b. d c. d self Check Find the distance on a number line between points with coordinates of a. 4 and 0 b. 22 and 27 Now Try Eercise 99. Self Check Answers. a. no b. {a, c, d, e, i, o, u} c. {a, e, i} 2. a. repeats b. terminates 3. a. 23,, b. 4, 6 c. " 4. a. Commutative Property of Multiplication b. Associative Property of Multiplication c. Commutative Property of Addition. 2 3/ , 4 ( 2 ] 8. 30, 34 [ ] `, 2 ) 9. a. 0 b. 2 c a. " 2 2 b a. 6 b.

13 Section A. Sets of Real Numbers A3 Eercises A. Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice eercises. 24. The between two distinct points on a number line is always positive. Let N the set of natural numbers Fill in the blanks. W the set of whole numbers. A is a collection of objects. Z the set of integers 2. If every member of one set B is also a member of a Q the set of rational numbers second set A, then B is called a of A. R the set of real numbers 3. If A and B are two sets, the set that contains all members that are in sets A and B or both is called Determine whether each statement is true or false. Read the of A and B. the symbol ( as is a subset of. 4. If A and B are two sets, the set that contains 2. N ( W 26. Q ( R all members that are in both sets is called the of A and B. 27. Q ( N 28. Z ( Q. A real number is any number that can be epressed 29. W ( Z 30. R ( Z as a. Practice 6. A is a letter that is used to represent a Let A a, b, c, d, e6, B d, e, f, g6, and number. C a, c, e, f6. Find each set. 7. The smallest prime number is. 3. A c B 32. A d B 8. All integers that are eactly divisible by 2 are called 33. A d C 34. B c C integers. 9. Natural numbers greater than that are not prime Determine whether the decimal form of each fraction are called numbers. terminates or repeats. 0. Fractions such as 2 3, 8 2, and 27 9 are called 9 numbers Irrational numbers are that don t terminate and don t repeat The symbol is read as is less than or equal to. 3. On a number line, the numbers are to the Consider the following set: left of 0. 2, 24, 2 2 3, 0,, "2, 2, 2.7, 6, The only integer that is neither positive nor negative 39. Which numbers are natural numbers? is. 40. Which numbers are whole numbers?. The Associative Property of Addition states that 4. Which numbers are integers? y2 z. 42. Which numbers are rational numbers? 6. The Commutative Property of Multiplication states that y. 7. Use the Distributive Property to complete the statement: 43. Which numbers are irrational numbers? m Which numbers are prime numbers? 8. The statement m n2p pm n2 illustrates the 4. Which numbers are composite numbers? Property of. 46. Which numbers are even integers? 9. The graph of an is a portion of a number 47. Which numbers are odd integers? line. 48. Which numbers are negative numbers? 20. The graph of an open interval has endpoints. 2. The graph of a closed interval has endpoints. Graph each subset of the real numbers on a number line. 22. The graph of a interval has one endpoint. 49. The natural numbers between and 23. Ecept for 0, Not the absolute value of every For number is Sale.

14 Not For Sale A4 Appendi A A Review of Basic Algebra 0. The composite numbers less than 0. The prime numbers between 0 and 20 Write each inequality as the union of two intervals and graph the result. 77., 22 or The integers from 2 to # 2 or The integers between and # 2 or $ 3 4. The even integers between 9 and. The odd integers between 6 and ,.7, and Write each inequality in interval notation and graph the interval , ,, ,, , #, , # 6. # 66. $ , # #, # # # # $ $ $ $ 22 Write each pair of inequalities as the intersection of two intervals and graph the result and, 4 80., 23 or $ Write each epression without using absolute value symbols p p 9. p 2 p 92. 2p 93. and $ and # and, and. 0 Find the distance between each pair of points on the number line and and and and 220 Applications 0. What subset of the real numbers would you use to describe the populations of several cities? 02. What subset of the real numbers would you use to describe the subdivisions of an inch on a ruler? 03. What subset of the real numbers would you use to report temperatures in several cities? 04. What subset of the real numbers would you use to describe the financial condition of a business? 74. $ 23 and, 6 7. $ 28 and # and # 7 Discovery and Writing 0. Eplain why 2 could be positive. 06. Eplain why every integer is a rational number. 07. Is the statement 0 ab 0 0 a 0? 0 b 0 always true? Eplain. a 08. Is the statement ` b ` 0 a 0 b 2 02 always true? 0 b 0 Eplain.

15 Section A.2 Integer Eponents and Scientific Notation A 09. Is the statement 0 a b 0 0 a 0 0 b 0 always true? Eplain.. Eplain why it is incorrect to write a, b. c if a, b and b. c. 0. Under what conditions will the statement given in Eercise 09 be true? 2. Eplain why 0 b 2 a 0 0 a 2 b 0. bioraven/shutterstock.com natural-number Eponents A.2 Integer Eponents and Scientific Notation In this section, we will learn to. Define natural-number eponents. 2. Apply the rules of eponents. 3. Apply the rules for order of operations to evaluate epressions. 4. Epress numbers in scientific notation.. Use scientific notation to simplify computations. The number of cells in the human body is approimated to be one hundred trillion or 00,000,000,000,000. One hundred trillion is # # # 02, where ten occurs fourteen times. Fourteen factors of ten can be written as 0 4. In this section, we will use integer eponents to represent repeated multiplication of numbers.. Define Natural-Number Eponents When two or more quantities are multiplied together, each quantity is called a factor of the product. The eponential epression 4 indicates that is to be used as a factor four times. 4 factors of 4??? In general, the following is true. For any natural number n, n factors of n??? c? In the eponential epression n, is called the base, and n is called the eponent or the power to which the base is raised. The epression n is called a power of. From the definition, we see that a natural-number eponent indicates how many times the base of an eponential epression is to be used as a factor in a product. If an eponent is, the is usually not written: EXAMPLE Using the Definition of Natural-Number Eponents Write each epression without using eponents: Not For Sale a. 4 2 b c. 2 3 d e. 3 4 f. 32 4

16 Not For Sale A6 Appendi A A Review of Basic Algebra solution In each case, we apply the definition of natural-number eponents. a ? 4 6 Read 4 2 as four squared. b Read as negative four squared. c Read 2 3 as the negative of five cubed. d Read 22 3 as negative five cubed. e ???? Read 3 4 as 3 times to the fourth power. f ???? Read 32 4 as 3 to the fourth power. self Check Accent on Technology Comment Comment To find powers on a scientific calculator use the y key. Write each epression without using eponents: a. 7 3 b c. a 3 d. a2 4 Now Try Eercise 9. Note the distinction between a n and a2 n : n factors of a n a???? c? Also note the distinction between 2 n and 22 n : n factors of 2 n 2??? c? 2 Using a Calculator to Find Powers n factors of a a2 n a2 a2 a2? c? a2 n factors of 2 22 n ? c? 22 We can use a graphing calculator to find powers of numbers. For eample, consider Input 2.3 and press the key. Input 3 and press ENTER. Figure A-6 We see that as the figure shows above. 2. Apply the Rules of Eponents We begin the review of the rules of eponents by considering the product m n. Since m indicates that is to be used as a factor m times, and since n indicates that is to be used as a factor n times, there are m n factors of in the product m n. m n factors of m factors of n factors of m n??? c????? c? mn

17 Section A.2 Integer Eponents and Scientific Notation A7 This suggests that to multiply eponential epressions with the same base, we keep the base and add the eponents. Product Rule for Eponents If m and n are natural numbers, then m n mn Comment The Product Rule applies to eponential epressions with the same base. A product of two powers with different bases, such as 4 y 3, cannot be simplified. To find another property of eponents, we consider the eponential epression m 2 n. In this epression, the eponent n indicates that m is to be used as a factor n times. This implies that is to be used as a factor mn times. mn factors of n factors of m m 2 n m 2 m 2 m 2? c? m 2 mn This suggests that to raise an eponential epression to a power, we keep the base and multiply the eponents. To raise a product to a power, we raise each factor to that power. n factors of y n factors of n factors of y y2 n y2 y2 y2? c? y2??? c? 2 y? y? y? c? y2 n y n To raise a fraction to a power, we raise both the numerator and the denominator to that power. If y 2 0, then n factors of y a y b n a y b a y b a y b? c? a y b n factors of? c? yyy? c? y n y n n factors of y The previous three results are called the Power Rules of Eponents. Power Rules of Eponents If m and n are natural numbers, then m 2 n mn y2 n n y n a y b n n y n 2 02 y EXAMPLE 2 Using Eponent Rules to Simplify Epressions with Natural-Number eponents Simplify: a. 7 b. 2 y 3 y c d e. a y 2b f. a 2 2 Not For y z 3 b Sale

18 Not For Sale A8 Appendi A A Review of Basic Algebra solution In each case, we will apply the appropriate rule of eponents. a b. 2 y 3 y 2 y 3 7 y 4 c ?9 36 d e. a y 2b y 2 2 y y f. a 2 2 y z 3 b y 2 z y 2 z z self Check 2 Zero Eponent negative Eponents Simplify: a. y b. a 2 a c d. a 3a3 b 2 c 3 Now Try Eercise b c 2 02 If we assume that the rules for natural-number eponents hold for eponents of 0, we can write 0 n 0n n n Since 0 n n, it follows that if 2 0, then If we assume that the rules for natural-number eponents hold for eponents that are negative integers, we can write 2n n 2nn However, we know that n? n 2 02? n n n n, and any nonzero number divided by itself is. Since 2n n? n n, it follows that 2n n If n is an integer and 2 0, then 2n and n 2n n Because of the previous definitions, all of the rules for natural-number eponents will hold for integer eponents. EXAMPLE 3 solution Simplifying Epressions with Integer Eponents Simplify and write all answers without using negative eponents: a b c. 24 d. 26 e. 23 f We will use the definitions of zero eponent and negative eponents to simplify each epression. a b c d. 26 6

19 Section A.2 Integer Eponents and Scientific Notation A9 e f self Check 3 Simplify and write all answers without using negative eponents: a. 7a 0 b. 3a 22 c. a 24 a 2 d. a 3 a Quotient Rule for Eponents EXAMPLE 4 solution Now Try Eercise 9. To develop the Quotient Rule for Eponents, we proceed as follows: m n m a nb m 2n m 2n2 m2n 2 02 This suggests that to divide two eponential epressions with the same nonzero base, we keep the base and subtract the eponent in the denominator from the eponent in the numerator. If m and n are integers, then m n m2n 2 02 Simplifying Epressions with Integer Eponents Simplify and write all answers without using negative eponents: a. 8 b We will apply the Product and Quotient Rules of Eponents. a b self Check 4 Simplify and write all answers without using negative eponents: a. 26 b Now Try Eercise 69. EXAMPLE Simplifying Epressions with Integer Eponents Simplify and write all answers without using negative eponents: a. a 3 y y 3b b. a 2n Not For y b Sale

20 Not For Sale A20 Appendi A A Review of Basic Algebra solution We will apply the appropriate rules of eponents. a. a 3 y y 3b y y y 0 self Check A Fraction to a Negative Power b. a y b 2n 2n y 2n y0 0 2n n y n y 2n n y n n y n Multiply numerator and denominator by in the form n y n. 0 y n y 0 n 2n n 0 and y 2n y n y 0. yn n 0 and y 0. a y n b Simplify and write all answers without using negative eponents: a. a 4 y y 2b 2 Now Try Eercise 7. b. a 2a 3b b 23 Part b of Eample establishes the following rule. If n is a natural number, then a 2n y b a y n b ( 2 0 and y 2 0) 3. apply the Rules for Order of Operations to Evaluate Epressions When several operations occur in an epression, we must perform the operations in the following order to get the correct result. strategy for Evaluating Epressions Using Order of Operations If an epression does not contain grouping symbols such as parentheses or brackets, follow these steps:. Find the values of any eponential epressions. 2. Perform all multiplications and/or divisions, working from left to right. 3. Perform all additions and/or subtractions, working from left to right. If an epression contains grouping symbols such as parentheses, brackets, or braces, use the rules above to perform the calculations within each pair of grouping symbols, working from the innermost pair to the outermost pair. In a fraction, simplify the numerator and the denominator of the fraction separately. Then simplify the fraction, if possible.

21 Section A.2 Integer Eponents and Scientific Notation A2 Comment Many students remember the Order of Operations Rule with the acronym PEMDAS: Parentheses Eponents Multiplication Division Addition Subtraction EXAMPLE 6 solution For eample, to simplify Evaluating Algebraic Epressions If 22, y 3, and z 24, evaluate a. 2 2 y 2 z b. 2z3 2 3y , we proceed as follows: Simplify within the inner parentheses: Simplify within the parentheses: , and Evaluate the power: In each part, we will substitute the numbers for the variables, apply the rules of order of operations, and simplify. a. 2 2 y 2 z Evaluate the powers Do the multiplication. 240 Do the addition. b. 2z3 2 3y Evaluate the powers. Do the multiplications. Do the subtraction Simplify the fraction. self Check 6 If 3 and y 22, evaluate y 2 2 y. Now Try Eercise Epress Numbers in Scientific Notation Not Scientists For often work with numbers Sale that are very large or very small. These numbers can be written compactly by epressing them in scientific notation.

22 Not For Sale A22 Appendi A A Review of Basic Algebra scientific Notation A number is written in scientific notation when it is written in the form N 3 0 n where # 0 N 0, 0 and n is an integer. ollirg/shutterstock.com EXAMPLE 7 solution Light travels 29,980,000,000 centimeters per second. To epress this number in scientific notation, we must write it as the product of a number between and 0 and some integer power of 0. The number lies between and 0. To get 29,980,000,000, the decimal point in must be moved ten places to the right. This is accomplished by multiplying by 0 0. Standard notation 29,980,000, Scientific notation One meter is approimately mile. To epress this number in scientific notation, we must write it as the product of a number between and 0 and some integer power of 0. The number 6.24 lies between and 0. To get , the decimal point in 6.24 must be moved four places to the left. This is accomplished by multiplying 6.24 by 0 4 or by multiplying 6.24 by 024. Standard notation Scientific notation To write each of the following numbers in scientific notation, we start to the right of the first nonzero digit and count to the decimal point. The eponent gives the number of places the decimal point moves, and the sign of the eponent indicates the direction in which it moves. a places to the right. b places to the left. c No movement of the decimal point. Writing Numbers in Scientific Notation Write each number in scientific notation: a. 62,000 b a. We must epress 62,000 as a product of a number between and 0 and some integer power of 0. This is accomplished by multiplying 6.2 by , b. We must epress as a product of a number whose absolute value is between and 0 and some integer power of 0. This is accomplished by multiplying 22.7 by self Check 7 Write each number in scientific notation: a. 293,000,000 b Now Try Eercise 03. EXAMPLE 8 Writing Numbers in Standard Notation Write each number in standard notation: a b

23 Section A.2 Integer Eponents and Scientific Notation A23 solution a. The factor of 0 2 indicates that 7.3 must be multiplied by 2 factors of 0. Because each multiplication by 0 moves the decimal point one place to the right, we have b. The factor of 0 2 indicates that 3.27 must be divided by factors of 0. Because each division by 0 moves the decimal point one place to the left, we have Accent on Technology self Check 8 Write each number in standard notation: a b example 9 solution Now Try Eercise.. Use Scientific Notation to Simplify Computations Another advantage of scientific notation becomes evident when we multiply and divide very large and very small numbers. Using Scientific Notation to Simplify Computations 3,400, Use scientific notation to calculate. 70,000,000 After changing each number to scientific notation, we can do the arithmetic on the numbers and the eponential epressions separately. 3,400, ,000, self Check 9 Use scientific notation to simplify Now Try Eercise 9. Scientific Notation 92, ,002 Graphing calculators will often give an answer in scientific notation. For eample, consider 2 8. Figure A-7 We see in the figure that the answer given is E0, which means Not For Sale

24 Not For Sale A24 Appendi A A Review of Basic Algebra Comment 2 8 To calculate an epression like on a scientific calculator, it is necessary to convert the denominator to scientific notation because the number has too many digits to fit on the screen. For scientific notation, we enter these numbers and press these keys: 6. EXP /2. To evaluate the epression above, we enter these numbers and press these keys: 2 y EXP /2 The display will read In standard notation, the answer is approimately 620,046,874,800,000,000,000. Self Check Answers. a. 7? 7? b c.? a? a? a d. a2 a2 a2 a2 62? a? a? a? a 2. a. y 6 b. a 8 c. 2 Eercises A.2 Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice eercises. Fill in the blanks.. Each quantity in a product is called a of the product. 2. A number eponent tells how many times a base is used as a factor. 3. In the epression 22 3, is the eponent and is the base. 4. The epression n is called an epression.. A number is in notation when it is written in the form N 3 0 n, where # 0 N 0, 0 and n is an. 6. Unless indicate otherwise, are performed before additions. Complete each eponent rule. Assume m n 8. m 2 n 9. y2 n 0. m n n Practice Write each number or epression without using eponents d. 27a9 b 6 c 9 3. a. 7 b. 3 a 2 c. a 2 d. a 2 4. a. 8 b.. a. 4 27b3 y 0 b. 8a a b a. 6,300 b Write each epression using eponents yyyy a2 2a2 2a t2 3t2 23t b2 2b2 2b2 2b2 29. yy 30. aaabbbb Use a calculator to simplify each epression Simplify each epression. Write all answers without using negative eponents. Assume that all variables are restricted to those numbers for which the epression is defined y 3 y z t y y a 3 a 6 2a 4 4. z z t t a a a a y y z 4 2 6

25 Section A.2 Integer Eponents and Scientific Notation A2 49. a a2 b b 3 0. a y 3b z t y 22 y m 22 m y 22 y a2 a a m3 3 n 2 b r 62. r 2 t3 64. t s9 s 3 s a t4 t 3b a aa 70. r9 r 23 2 r a a23 b 2b a r4 r 26 r 3 r 23b 2 7. a y y 2b a 23 y y 23 b a 3 y y 3b z 23 y2 2 y 2 z yz m22 n 3 p mn 22 p mn 22 p mn 2 p a t24 t 23b a 27 y 7 y 24b a 32 y y 26b a 224 y 3 z y 23 z b 3 Simplify each epression Let 22, y 0, and z 3 and evaluate each epression z z z 3 2 z 2 2 y 92. z2 2 2 y z y 3 z z2 2 2y 2 z2 3 Epress each number in scientific notation , , ,000, ,470,000, one trillion 06. one millionth Epress each number in standard notation , Use the method of Eample 9 to do each calculation. Write all answers in scientific notation. 6,0002 4, , ,000, , , ,000 4,000,000, , , ,000,000,000 applications Use scientific notation to compute each answer. Write all answers in scientific notation. 2. Speed of sound The speed of sound in air is centimeters per second. Compute the speed of sound in meters per minute. 22. Volume of a bo Calculate the volume of a bo that has dimensions of 6,000 by 9,700 by 4,700 millimeters. 23. Mass of a proton The mass of one proton is gram. Find the mass of one billion protons z z 23 2 Not For 2 Sale 6 2 z 23 z 22

26 Not For Sale A26 Appendi A A Review of Basic Algebra 24. Speed of light The speed of light in a vacuum is approimately 30,000,000,000 centimeters per second. Find the speed of light in miles per hour. 60,934.4 cm mile.2 2. Astronomy The distance d, in miles, of the nth planet from the sun is given by the formula d 9,27, n To the nearest million miles, find the distance of Earth and the distance of Mars from the sun. Give each answer in scientific notation. Mercury Jupiter Mars Earth Venus 26. License Plates The number of different license plates of the form three digits followed by three letters, as in the illustration, is 0? 0? 0? 26? 26? 26. Write this epression using eponents. Then evaluate it and epress the result in scientific notation. UTAH WB 23ABC COUNTY 2 discovery and Writing 27. New way to the center of the earth The spectacular blue marble image is the most detailed true-color image of the entire Earth to date. A new NASAdeveloped technique estimates Earth s center of mass within millimeter (0.04 inch) a year by using a combination of four space-based techniques. The distance from the Earth s center to the North Pole (the polar radius) measures approimately 6,36.70 km, and the distance from the center to the equator (the equatorial radius ) measures approimately 6,378.3 km. Epress each distance using scientific notation. 28. Refer to Eercise 27. Given that km is approimately equal to 0.62 miles, use scientific notation to epress each distance in miles. Write each epression with a single base. 29. n m 3 3. m 2 3m m a n23 a 3 3. Eplain why 2 4 and 22 4 represent different numbers. 36. Eplain why is not in scientific notation. Review 37. Graph the interval 22, Graph the interval 2`,234 c 33,` Evaluate 0 p Find the distance between 27 and 2 on the number line. NASA Goddard Space Flight Center

27 Section A.3 Rational Eponents and Radicals A27 A.3 Rational Eponents and Radicals In this section, we will learn to. Define rational eponents whose numerators are. 2. Define rational eponents whose numerators are not. 3. Define radical epressions. 4. Simplify and combine radicals.. Rationalize denominators and numerators. Medioimages/Photodisc/Jupiter Images Caution In the epression a /n, there is no real-number nth root of a when n is even and a, 0. For eample, 2642 /2 is not a real number, because the square of no real number is 264. rational Eponents Dead Man s Curve is a 964 hit song by the rock and roll duo Jan Berry and Dean Torrence. The song details a teenage drag race that ends in an accident. Today, dead man s curve is a commonly used epression given to dangerous curves on our roads. Every curve has a critical speed. If we eceed this speed, regardless of how skilled a driver we are, we will lose control of the vehicle. The radical epression 3.9"r gives the critical speed in miles per hour when we travel a curved road with a radius of r feet. A knowledge of square roots and radicals is important and used in the construction of safe highways and roads. We will study the topic of radicals in this section.. Define Rational Eponents Whose Numerators Are If we apply the rule m 2 n mn to 2 /2 2 2, we obtain 2 / /222 Keep the base and multiply the eponents. 2 2? 2 2 Thus, 2 /2 is a real number whose square is 2. Although both 2 2 and , we define 2 /2 to be the positive real number whose square is 2: 2 /2 Read 2 /2 as the square root of 2. In general, we have the following definition. If a $ 0 and n is a natural number, then a /n (read as the nth root of a ) is the nonnegative real number b such that b n a Since b a /n, we have b n a /n 2 n a. EXAMPLE Simplifying Epressions with Rational Eponents In each case, we will apply the definition of rational eponents. a. 6 /2 4 Because Read 6 /2 as the square root of 6. b. 27 /3 3 Because Read 27 /3 as the cube root of 27. c. a /4 8 b Because a b 8. Read a /4 Not For Sale 8 b as the fourth root of 8.

28 Not For Sale A28 Appendi A A Review of Basic Algebra d. 232 / 232 / 2 Read 32 / as the fifth root of Because self Check Simplify: a. 00 /2 b. 243 / now Try Eercise. EXAMPLE 2 If n is even in the epression a /n and the base contains variables, we often use absolute value symbols to guarantee that an even root is nonnegative / / /6 3 2 Because Since could be negative, absolute value symbols are necessary to guarantee that the square root is nonnegative. Because Since could be negative, absolute value symbols are necessary to guarantee that the fourth root is nonnegative. Because Since 2 is always nonnegative, no absolute value symbols are necessary. Read /6 as the sith root of If n is an odd number in the epression a /n, the base a can be negative. Simplifying Epressions with Rational Eponents Simplify by using the definition of rational eponent. a. 282 /3 22 Because b. 23,22 / 2 Because 22 23,2. c. a2,000 b /3 2 0 Because a2 0 b 3 2,000. self Check 2 Simplify: a. 222 /3 b. 200,0002 / Caution now Try Eercise 9. If n is odd in the epression a /n, we don t need to use absolute value symbols, because odd roots can be negative /3 23 Because a 7 2 /7 22a Because 22a a 7. We summarize the definitions concerning a /n as follows. summary of Definitions of a /n If n is a natural number and a is a real number in the epression a /n, then If a $ 0, then a /n is the nonnegative real number b such that b n a. If a, 0 e and n is odd, then a/n is the real number b such that b n a. and n is even, then a /n is not a real number. The following chart also shows the possibilities that can occur when simplifying a /n.

29 Section A.3 Rational Eponents and Radicals A29 Strategy for Simplifying Epressions of the Form a /n Rule for Rational Eponents a 0 a. 0 a, 0 a, 0 a n a /n Eamples n is a natural number. n is a natural number. n is an odd natural number. n is an even natural number. 0 /n is the real number 0 because 0 n 0. a /n is the nonnegative real number such that a /n 2 n a. a /n is the real number such that a /n 2 n a. a /n is not a real number. 0 /2 0 because / 0 because /2 4 because /3 3 because / 22 because /3 2 because /2 is not a real number. 282 /4 is not a real number. 2. define Rational Eponents Whose Rational Eponents Are Not The definition of a /n can be etended to include rational eponents whose numerators are not. For eample, 4 3/2 can be written as either 4 /2 2 3 or /2 Because of the Power Rule, m 2 n mn. This suggests the following rule. If m and n are positive integers, the fraction m n is in lowest terms, and a/n is a real number, then a m/n a /n 2 m a m 2 /n In the previous rule, we can view the epression a m/n in two ways:. a /n 2 m : the mth power of the nth root of a 2. a m 2 /n : the nth root of the mth power of a For eample, /3 can be simplified in two ways: / /3 4 2 or / / /3 9 9 Not As For this eample suggests, Sale it is usually easier to take the root of the base first to avoid large numbers.

30 Not For Sale A30 Appendi A A Review of Basic Algebra Comment It is helpful to think of the phrase power over root when we see a rational eponent. The numerator of the fraction represents the power and the denominator represents the root. Begin with the root when simplifying to avoid large numbers. Negative Rational Eponents If m and n are positive integers, the fraction m n is in lowest terms and a/n is a real number, then a 2m/n a and m/n a 2m/n am/n a 2 02 EXAMPLE 3 Simplifying Epressions with Rational Eponents We will apply the rules for rational eponents. a. 2 3/2 2 /2 2 3 b. a2 6 2/3,000 b c a2 6 /3 2,000 b d 3 a2 2 0 b c / 32 d. 2/ 83/4 23/ / 2 2 8/ Self Check 3 Simplify: a. 49 3/2 b. 6 23/4 c. Now Try Eercise /3 Because of the definition, rational eponents follow the same rules as integer eponents. EXAMPLE 4 Using Eponent Rules to Simplify Epressions with Rational Eponents Simplify each epression. Assume that all variables represent positive numbers, and write answers without using negative eponents. a. 362 /2 36 /2 /2 b. a/3 b 2/3 2 6 y a6/3 b 2/3 y 6 6 /2 a2 b 4 y 6

31 Section A.3 Rational Eponents and Radicals A3 c. a/2 a /4 a /2/42/6 d. c 2c22/ /3 d 2c 22/24/ 2 /3 a /6 c 4/ a 6/23/222/ c 26/ 2 4 /3 a 7/2 22 /3 c 26/ 2 /3 2c 230/ 2c 22 2 c 2 Definition of " n a nth Root of a Nonnegative Number Self Check 4 Use the directions for Eample 4: a. a y2 /2 49 b b. b3/7 b 2/7 c. 9r2 s2 /2 b 4/7 rs 23/2 Now Try Eercise Define Radical Epressions Radical signs can also be used to epress roots of numbers. If n is a natural number greater than and if a /n is a real number, then " n a a /n In the radical epression " n a, the symbol " is the radical sign, a is the radicand, and n is the inde (or the order) of the radical epression. If the order is 2, the epression is a square root, and we do not write the inde. "a " 2 a If the inde of a radical is 3, we call the radical a cube root. If n is a natural number greater than and a $ 0, then " n a is the nonnegative number whose nth power is a. Q" n ar n a Caution In the epression " n a, there is no real-number nth root of a when n is even and a, 0. For eample, "264 is not a real number, because the square of no real number is 264. If 2 is substituted for n in the equation Q" n ar n a, we have Q" 2 ar 2 Q"aR 2 "a"a a for a $ 0 This shows that if a number a can be factored into two equal factors, either of those factors is a square root of a. Furthermore, if a can be factored into n equal factors, any one of those factors is an nth root of a. If n is an odd number greater than in the epression " n a, the radicand can be negative. Not For Sale

32 Not For Sale A32 Appendi A A Review of Basic Algebra EXAMPLE Finding nth Roots of Real Numbers We apply the definitions of cube root and fifth root. a. " Because b. " Because c. Å , Because a2 3 0 b ,000. summary of Definitions of " n a strategy for Simplifying Epressions of the Form " n a d. 2" Q" 2243R self Check Find each root: a. " 3 26 b. Å 2 32 Now Try Eercise 69. We summarize the definitions concerning " n a as follows. If n is a natural number greater than and a is a real number, then The following chart also shows the possibilities that can occur when simplifying " n a. a 0 a. 0 If a $ 0, then " n a is the nonnegative real number such that Q" n ar n a. If a, 0 e and n is odd, then "n a is the real number such that Q" n ar n a. and n is even, then " n a is not a real number. a n " n a Eamples n is a natural number greater than. n is a natural number greater than. " n 0 is the real number 0 because 0 n 0. " n a is the nonnegative real number such that Q" n ar n a. " because " 0 0 because 0 0. "6 4 because " because a, 0 a, 0 n is an odd natural number greater than. n is an even natural number. " n a is the real number such that Q" n ar n a. " n a is not a real number. " because " because "29 is not a real number. " 4 28 is not a real number.

33 Section A.3 Rational Eponents and Radicals A33 We have seen that if a /n is real, then a m/n a /n 2 m a m 2 /n. This same fact can be stated in radical notation. a m/n " n a2 m " n a m Thus, the mth power of the nth root of a is the same as the nth root of the mth power of a. For eample, to find " , we can proceed in either of two ways: " Q" 3 27R or " " EXAMPLE 6 solution By definition, "a 2 represents a nonnegative number. If a could be negative, we must use absolute value symbols to guarantee that "a 2 will be nonnegative. Thus, if a is unrestricted, "a 2 0 a 0 A similar argument holds when the inde is any even natural number. The symbol " 4 a 4, for eample, means the positive fourth root of a 4. Thus, if a is unrestricted, " 4 a 4 0 a 0 Simplifying Radical Epressions If is unrestricted, simplify a. " b. " 3 3 c. "9 8 We apply the definitions of sith roots, cube roots, and square roots. a. " Use absolute value symbols to guarantee that the result will be nonnegative. b. " 3 3 Because the inde is odd, no absolute value symbols are needed. c. " Because 3 4 is always nonnegative, no absolute value symbols are needed. self Check 6 Use the directions for Eample 6: a. " b. " 3 27y 3 c. " 4 8 Now Try Eercise Simplify and Combine Radicals Many properties of eponents have counterparts in radical notation. For eample, since a /n b /n ab2 /n and a/n b /n a b 2 /n and b 2 02, we have the following. Multiplication and Division Properties of Radicals If all epressions represent real numbers, " n a" n b " n " n a ab " n b n a Åb b 2 02 In words, we say The product of two nth roots is equal to the nth root of their product. Not For Sale The quotient of two nth roots is equal to the nth root of their quotient.

34 Not For Sale A34 Appendi A A Review of Basic Algebra Caution These properties involve the nth root of the product of two numbers or the nth root of the quotient of two numbers. There is no such property for sums or differences. For eample, "9 4 2 "9 "4, because "9 4 "3 but "9 "4 3 2 and "3 2. In general, "a b "a "b and "a 2 b "a 2 "b Numbers that are squares of positive integers, such as, 4, 9, 6, 2, and 36 are called perfect squares. Epressions such as 4 2 and 9 6 are also perfect squares, because each one is the square of another epression with integer eponents and rational coefficients and 9 6 a b Numbers that are cubes of positive integers, such as, 8, 27, 64, 2, and 26 are called perfect cubes. Epressions such as 64 3 and 27 9 are also perfect cubes, because each one is the cube of another epression with integer eponents and rational coefficients and 27 9 a b There are also perfect fourth powers, perfect fifth powers, and so on. We can use perfect powers and the Multiplication Property of Radicals to simplify many radical epressions. For eample, to simplify "2, we factor 2 so that one factor is the largest perfect square that divides 2. In this case, it is 4 4. We then rewrite 2 as 4 4? 3 and simplify. "2 "4 4? 3 Factor 2 as 4 4? 3. "4 4 "3 Use the Multiplication Property of Radicals: "ab "a"b. 2 2 "3 " To simplify " y, we find the largest perfect-cube factor of y (which is 26 9 ) and proceed as follows: " y " ? 2y Factor y as 26 9? 2y. " " 3 2y Use the Multiplication Property of Radicals: " 3 ab " 3 a" 3 b. 6 3 " 3 2y " Radical epressions with the same inde and the same radicand are called like or similar radicals. We can combine the like radicals in 3"2 2"2 by using the Distributive Property. 3"2 2"2 3 22"2 "2

35 Section A.3 Rational Eponents and Radicals A3 This eample suggests that to combine like radicals, we add their numerical coefficients and keep the same radical. When radicals have the same inde but different radicands, we can often change them to equivalent forms having the same radicand. We can then combine them. For eample, to simplify "27 2 "2, we simplify both radicals and combine like radicals. "27 2 "2 "9? 3 2 "4? 3 Factor 27 and 2. "9"3 2 "4"3 "ab "a"b 3"3 2 2"3 "9 3 and "4 2. "3 Combine like radicals. EXAMPLE 7 Adding and Subtracting Radical Epressions Simplify: a. "0 "200 b. 3z" 64z 2 2" 2z 6 solution We will simplify each radical epression and then combine like radicals. a. "0 "200 "2? 2 "00? 2 "2"2 "00"2 "2 0"2 "2 b. 3z" 64z 2 2" 2z 6 3z" 32? 2z 2 2" z? 2z 3z" 32 " 2z 2 2" z " 2z 3z22" 2z 2 2z" 2z 6z" 2z 2 2z" 2z 4z" 2z self Check 7 Simplify: a. "8 2 "8 b. 2" 3 8a 4 a" 3 24a now Try Eercise 8.. Rationalize Denominators and Numerators By rationalizing the denominator, we can write a fraction such as " "3 as a fraction with a rational number in the denominator. All that we must do is multiply both the numerator and the denominator by "3. (Note that "3"3 is the rational number 3.) " "3 ""3 "3"3 " 3 To rationalize the numerator, we multiply both the numerator and the denominator by ". (Note that "" is the rational number.) " Not For "3 "" "3" Sale "

36 Not For Sale A36 Appendi A A Review of Basic Algebra EXAMPLE 8 Rationalizing the Denominator of a Radical Epression Rationalize each denominator and simplify. Assume that all variables represent positive numbers. a. "7 b. Å c. Å 3 d. Å 3a 3 solution We will multiply both the numerator and the denominator by a radical that will make the denominator a rational number. a. "7 "7 "7"7 "7 7 c. Å 3 "3 " "3" "" "3 self Check 8 Use the directions for Eample 8: a. b. "6 Å Now Try Eercise 0. b. Å "3 3 " 3 4 "3 3" 3 2 " 3 4" 3 2 "3 6 " 3 8 "3 6 2 d. Å 3a 3 "3a3 " "3a3 " " " "a3 "2 6 "a2 "a 3 a"a 3 Multiply numerator and denominator by " 3 2, because " 3 4" 3 2 " Multiply numerator and denominator by ", because " " " EXAMPLE 9 solution Rationalizing the Numerator of a Radical Epression Rationalize each numerator and simplify. Assume that all variables represent positive numbers: a. " b. 2" We will multiply both the numerator and the denominator by a radical that will make the numerator a rational number.

37 Section A.3 Rational Eponents and Radicals A37 a. " 7 "? " 7" 7" b. 2" "3 9? " " " " " " Divide out the 3 s. self Check 9 Use the directions for Eample 9: EXAMPLE 0 solution a. "2 b. 3"3 2y 2 6 Now Try Eercise. After rationalizing denominators, we often can simplify an epression. Rationalizing Denominators and Simplifying Simplify: Å 2 Å 8. We will rationalize the denominators of each radical and then combine like radicals. Å 2 Å 8 "2 "8 Å 2 " "2 "2 "2"2 "2 "8"2 "2 2 "2 "6 "2 2 "2 4 3"2 4 ; "2 Å 8 " "8 "8 self Check 0 3 Simplify: Å Å6. now Try Eercise. Another property of radicals can be derived from the properties of eponents. If all of the epressions represent real numbers, # n " m " n /m /m 2 /n / mn2 mn " m n #" m " /n /n 2 /m /nm2 mn " These results are summarized in the following theorem (a fact that can be proved). Not For Sale

38 Not For Sale A38 Appendi A A Review of Basic Algebra Theorem If all of the epressions involved represent real numbers, then m #" n # n " m mn " We can use the previous theorem to simplify many radicals. For eample, # 3 "8 #" 3 8 "2 Rational eponents can be used to simplify many radical epressions, as shown in the following eample. EXAMPLE solution Simplifying Radicals Using the Previously Stated Theorem Simplify. Assume that and y are positive numbers. a. " 6 4 b. " 2 3 c. " 9 8y 3 In each case, we will write the radical as an eponential epression, simplify the resulting epression, and write the final result as a radical. a. " / /6 2 2/6 2 /3 " 3 2 b. 2 " 3 3/2 /4 " 4 c. " 9 8y y 3 2 /9 2y2 3/9 2y2 /3 " 3 2y self Check Simplify: a. " 4 4 b. " Now Try Eercise 7. Self Check Answers. a. 0 b a. 2 b a. 343 b. 8 c a. y b. b /7 c. 3s 2. a. 6 b a b. 3y c a. "2 b. 8a" 3 3a 8. a. "6 b. " y 9. a. b. 0. "3 4. a. "2 b. " 3 3 "2 " 3 4y 4 Eercises A.3 Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice eercises. Fill in the blanks.. If a 0 and n is a natural number, then a /n. 2. If a. 0 and n is a natural number, then a /n is a number. 3. If a, 0 and n is an even number, then a /n is a real number /3 can be written as or.. " n a 6. "a 2 7. " n a" n b n a 8. Å b 9. " y " "y 0. m #" n or # n " m can be written as.

39 Section A.3 Rational Eponents and Radicals A39 Practice Simplify each epression.. 9 / /3 3. a 2 b /2 4. a 6 62 b /4. 28 /4 6. 2a 8 27 b /3 7. 0,0002 /4 8.,024 / 9. a b / / / /3 Simplify each epression. Use absolute value symbols when necessary a 2 2 / a 4 2 /2 2. 6a 4 2 / a 3 2 / a 2 / a 6 2 / b 6 2 / t 8 2 /4 3. a 6a4 /2 2b 2b 33. a2,0006 /3 27y 3 b 32. a2 a 32b 0b / 34. a 49t2 00z 4b /2 Simplify each epression. Write all answers without using negative eponents / / / / ,000 2/ / / / / / / /3 47. a 4 9 b /2 49. a b 22/3 48. a 2 8 b 3/2 0. a 2 8 b 24/3 Simplify each epression. Assume that all variables represent positive numbers. Write all answers without using negative eponents.. 00s 4 2 / u 6 v 3 2 / y 0 z 2 2/ 4. 62a 4 b 8 2 2/4. 0 y 2 3/ 6. 64a 6 b 2 2 /6 6. a 27r6,000s 2b 22/3 7. r 8 s / y / a2 8a6 2/3 2b 9b 60. a 64 3/4 Not 62y 8b For Sale 3 2u 4 0. Å 9v 62. a2 32m0 243n b 22/ 63. a2/ a 4/ 64. 6/7 3/7 a / 2/7 /7 Simplify each radical epression. 6. " "8 67. " " " " Å , Å Simplify each epression, using absolute value symbols when necessary. Write answers without using negative eponents. 73. " "2y 2 7. "9y "a 4 b " 3 8y " 3 227z Å 4 4 y 8 z a 0 b Å c Simplify each epression. Assume that all variables represent positive numbers so that no absolute value symbols are needed. 8. "8 2 "2 82. "7 2 2" "200 2 " "28a 3 2 a"62a 8. 2"48y 2 3y"2y y"2y 4"7y " 3 8 3" " " " 4 768z " 4 48z " 64y 2 3" 486y 2 9. "8 2 y 2 "2y "0 2 y 92. 3"8 2"2 3 2 " " 3 6y 4 y" 3 2y 2 " 3 4y " " 4 32 " 4,20 Rationalize each denominator and simplify. Assume that all variables represent positive numbers "3 " " "y 2 4d " 3 2 " 3 9 a " 3 2a " 3 36c 2b 04. " 4 3a 2 Å 2y 06. Å 3 2 3s 4r 2

40 Not For Sale A40 Appendi A A Review of Basic Algebra Rationalize each numerator and simplify. Assume that all variables are positive numbers. 07. " " " 6b 3 64a 08. "y 3 0. "3 6b Å 7 Use this idea to write each of the following epressions as a single radical. 2. "2" "3" " "3 2 "2 " 2. For what values of does " 4 4? Eplain. 26. If all of the radicals involved represent real numbers and y 2 0, eplain why Rationalize each denominator and simplify. 3. Å Å 27 Å 2 3 Å6. Å 8 2 Å 2 Å 32 Simplify each radical epression. 7. " " " " y 3 y 3 y 6. Å 4 Å32 2 Å00 discovery and Writing We often can multiply and divide radicals with different indices. For eample, to multiply "3 by " 3, we first write each radical as a sith root "3 3 /2 3 3/6 " " 6 27 " 3 /3 2/6 " 6 2 " 6 2 and then multiply the sith roots. "3" 3 " 6 27" 6 2 " " 6 67 Division is similar. A.4 Polynomials In this section, we will learn to. Define polynomials. 2. Add and subtract polynomials. 3. Multiply polynomials. 4. Rationalize denominators.. Divide polynomials. n Å y "n " n y 27. If all of the radicals involved represent real numbers and there is no division by 0, eplain why a 2m/n y b n y m Å m 28. The definition of m/n requires that " n be a real number. Eplain why this is important. (Hint: Consider what happens when n is even, m is odd, and is negative.) Review 29. Write 22, # using interval notation. 30. Write the epression without using absolute value symbols. Assume that. 4. Evaluate each epression when 22 and y y 2 y 4y Write 67,000,000 in scientific notation. 34. Write in standard notation. PCN Photography/Alamy Football is one of the most popular sports in the United States. Brett Favre, one of the most talented quarterbacks ever to play the game, holds the record for the most career NFL touchdown passes, the most NFL pass completions, and the most passing yards. He led the Green Bay Packers to the Super Bowl and most recently played for the Minnesota Vikings. His nickname is Gunslinger.

41 Section A.4 Polynomials A4 An algebraic epression can be used to model the trajectory or path of a football when passed by Brett Favre. Suppose the height in feet, t seconds after the football leaves Brett s hand, is given by the algebraic epression 20.t 2 t. At a time of t 3 seconds, we see that the height of the football is ft. Algebraic epressions like 20.t 2 t. are called polynomials, and we will study them in this section.. Define Polynomials A monomial is a real number or the product of a real number and one or more variables with whole-number eponents. The number is called the coefficient of the variables. Some eamples of monomials are 3, 7ab 2, 2ab 2 c 4, 3, and 22 with coefficients of 3, 7, 2,, and 22, respectively. The degree of a monomial is the sum of the eponents of its variables. All nonzero constants (ecept 0) have a degree of 0. The degree of 3 is. The degree of 7ab 2 is 3. The degree of 2ab 2 c 4 is 7. The degree of 3 is 3. The degree of 22 is 0 (since ). 0 has no defined degree. A monomial or a sum of monomials is called a polynomial. Each monomial in that sum is called a term of the polynomial. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial. Monomials Binomials Trinomials 3 2 2a 3b y y 4 2 2y 2 a 2 b 3 c y y 2 2 8y 2 24 The degree of a polynomial is the degree of the term in the polynomial with highest degree. The only polynomial with no defined degree is 0, which is called the zero polynomial. Here are some eamples. 3 2 y 3 y 2 7 is a trinomial of th degree, because its term with highest degree (the first term) is. 3ab a 2 b is a binomial of degree 3. 3y 2 " 4 3z 4 2 "7 is a polynomial, because its variables have wholenumber eponents. It is of degree 4. 27y /2 3y 2 " 3z is not a polynomial, because one of its variables (y in the first term) does not have a whole-number eponent. If two terms of a polynomial have the same variables with the same eponents, they are like or similar terms. To combine the like terms in the sum 3 2 y 2 y or the difference 7y 2 2 2y 2, we use the Distributive Property: 3 2 y 2 y y 7y 2 2 2y y y y 2 Not This For illustrates that to combine Sale like terms, we add (or subtract) their coefficients and keep the same variables and the same eponents.

42 Not For Sale A42 Appendi A A Review of Basic Algebra 2. Add and Subtract Polynomials Recall that we can use the Distributive Property to remove parentheses enclosing the terms of a polynomial. When the sign preceding the parentheses is +, we simply drop the parentheses: EXAMPLE solution a b 2 c2 a b 2 c2 a b 2 c a b 2 c When the sign preceding the parentheses is 2, we drop the parentheses and the 2 sign and change the sign of each term within the parentheses. 2a b 2 c2 2a b 2 c2 2a 22b 2 22c 2a 2 b c We can use these facts to add and subtract polynomials. To add (or subtract) polynomials, we remove parentheses (if necessary) and combine like terms. Adding Polynomials Add: 3 3 y 2 2 2y2 2 3 y To add the polynomials, we remove parentheses and combine like terms. 3 3 y 2 2 2y2 2 3 y y 2 2 2y 2 3 y y 2 3 y y 3 Use the Commutative Property to rearrange terms. 3 y 2 2y 3 Combine like terms. We can add the polynomials in a vertical format by writing like terms in a column and adding the like terms, column by column. 3 3 y 2 2 2y 2 3 y y 2 2y 3 self Check Add: Now Try Eercise 2. EXAMPLE 2 solution Subtracting Polynomials Subtract: 2 2 3y y To subtract the polynomials, we remove parentheses and combine like terms y y y y y 2 2y Use the Commutative Property to rearrange terms. 2 y Combine like terms.

43 Section A.4 Polynomials A43 We can subtract the polynomials in a vertical format by writing like terms in a column and subtracting the like terms, column by column y y y y 2 3y 2 2y y y self Check 2 Subtract: Now Try Eercise 23. EXAMPLE 3 We can also use the Distributive Property to remove parentheses enclosing several terms that are multiplied by a constant. For eample, This eample suggests that to add multiples of one polynomial to another, or to subtract multiples of one polynomial from another, we remove parentheses and combine like terms. Using the Distributive Property and Combining Like Terms Simplify: 72y y solution 72y y y y Use the Distributive Property to remove parentheses. 4y 2 2 y Use the Commutative Property to rearrange terms. 9y Combine like terms. self Check 3 Simplify: 32b 2 2 3a 2 b2 2bb a 2 2. Now Try Eercise Multiply Polynomials To find the product of 3 2 y 3 z and yz 2, we proceed as follows: 3 2 y 3 z2 yz 2 2 3? 2? y 3? z??? y? z 2 3?? 2?? y 3? y? z? z 2 Use the Commutative Property to rearrange terms. 3 y 4 z 3 This illustrates that to multiply two monomials, we multiply the coefficients and then multiply the variables. To find the product of a monomial and a polynomial, we use the Distributive Property. 3y 2 2y 2 2 7yz2 3y 2 2y2 3y y 2 2 7yz2 6 2 y y 2 2 2y 3 z Not This For illustrates that to multiply Sale a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

44 Not For Sale A44 Appendi A A Review of Basic Algebra To multiply one binomial by another, we use the Distributive Property twice. EXAMPLE 4 Multiplying Binomials Multiply: a. y2 y2 b. 2 y2 2 y2 c. y2 2 y2 solution Special Product Formulas a. y2 y2 y2 y2y 2 y y y 2 2 2y y 2 b. 2 y2 2 y2 2 y2 2 2 y2y 2 2 y 2 y y y y 2 c. y2 2 y2 y2 2 y2y 2 y 2 y 2 y y 2 self Check 4 Multiply: a b c Now Try Eercise 4. The products in Eample 4 are called special products. Because they occur so often, it is worthwhile to learn their forms. y2 2 y2 y2 2 2y y 2 2 y2 2 2 y2 2 y y y 2 y2 2 y2 2 2 y 2 Caution Remember that y2 2 and 2 y2 2 have trinomials for their products and that y y 2 and 2 y y 2 For eample, and We can use the FOIL method to multiply one binomial by another. FOIL is an acronym for First terms, Outer terms, Inner terms, and Last terms. To use this method to multiply by 2, we write

45 Section A.4 Polynomials A4 First terms Last terms EXAMPLE solution self Check EXAMPLE 6 Inner terms Outer terms In this eample, the product of the first terms is 6 2, the product of the outer terms is, the product of the inner terms is 28, and the product of the last terms is 220. The resulting like terms of the product are then combined. Using the FOIL Method to Multiply Polynomials Use the FOIL method to multiply: Q"3 RQ2 2 "3R. " "32 2"3 2 "3"3 2 2 "3 2" "3 2 2"3 2 2 "3 2 Multiply: Q2 "3RQ 2 "3R. Now Try Eercise 63. To multiply a polynomial with more than two terms by another polynomial, we multiply each term of one polynomial by each term of the other polynomial and combine like terms whenever possible. Multiplying Polynomials Multiply: a. y2 2 2 y y 2 2 b solution a. y2 2 2 y y y y 2 y 2 2 y 2 y 3 b y self Check 6 Multiply: now Try Eercise 67. We can use a vertical format to multiply two polynomials, such as the polynomials given in Self Check 6. We first write the polynomials as follows and draw a line beneath them. We then multiply each term of the upper polynomial by each Not term of For the lower polynomial Sale and write the results so that like terms appear in each column. Finally, we combine like terms column by column.

46 Not For Sale A46 Appendi A A Review of Basic Algebra Multiply by Multiply by In each column, combine like terms. If n is a whole number, the epressions a n and 2a n 2 3 are polynomials and we can multiply them as follows: a n 2 2a n a 2n 2 3a n 2a n 2 3 2a 2n 2 a n 2 3 Combine like terms. We can also use the methods previously discussed to multiply epressions that are not polynomials, such as 22 y and 2 2 y y2 2 2 y y 2 2 y 2 y y 2 y 2 y y 2 y 2 0 and y 0. 2 y 2 2 y 4. Rationalize Denominators If the denominator of a fraction is a binomial containing square roots, we can use the product formula y2 2 y2 to rationalize the denominator. For eample, to rationalize the denominator of 6 "7 2 To rationalize a denominator means to change the denominator into a rational number. we multiply the numerator and denominator by "7 2 2 and simplify. 6 "7 2 6" "7 22 " " " " "7 2 2 "7 2 2 Here the denominator is a rational number. In this eample, we multiplied both the numerator and the denominator of the given fraction by " This binomial is the same as the denominator of the given fraction "7 2, ecept for the sign between the terms. Such binomials are called conjugate binomials or radical conjugates. Conjugate Binomials Conjugate binomials are binomials that are the same ecept for the sign between their terms. The conjugate of a b is a 2 b, and the conjugate of a 2 b is a b.

47 Section A.4 Polynomials A47 EXAMPLE 7 solution Rationalizing the Denominator of a Radical Epression "3 2 "2 Rationalize the denominator:. 02. "3 "2 We multiply the numerator and the denominator by "3 2 "2 (the conjugate of "3 "22 and simplify. "3 2 "2 Q"3 2 "2RQ"3 2 "2R "3 "2 Q"3 "2RQ"3 2 "2R "3 2 "2 "3 2 "2 "3"3 2 "3"2 2 "2"3 "2"2 Q"3R 2 2 Q"2R "6 2 " " self Check 7 Rationalize the denominator: " 2 " 2 2. EXAMPLE 8 solution Now Try Eercise 89. In calculus, we often rationalize a numerator. Rationalizing the Numerator of a Radical Epression Rationalize the numerator: " h 2 ". h To rid the numerator of radicals, we multiply the numerator and the denominator by the conjugate of the numerator and simplify. " h 2 " h Q" h 2 "RQ" h "R " h " hq" h "R " h " h 2 Here the numerator has no radicals. hq" h "R h hq" h "R " h " Divide out the common factor of h. self Check 8 Rationalize the numerator: "4 h 2 2. h Not For Sale Now Try Eercise 99.

48 Not For Sale A48 Appendi A A Review of Basic Algebra. Divide Polynomials To divide monomials, we write the quotient as a fraction and simplify by using the rules of eponents. For eample, 6 2 y y y y 2 2 3y2 To divide a polynomial by a monomial, we write the quotient as a fraction, write the fraction as a sum of separate fractions, and simplify each one. For eample, to divide 8 y y y 3 by 4 3 y 4, we proceed as follows: 8 y y y y 4 8 y y 4 22 y 4 3 y y y y 2 4 y To divide two polynomials, we can use long division. To illustrate, we consider the division which can be written in long division form as 7q The binomial 7 is called the divisor, and the trinomial is called the dividend. The final answer, called the quotient, will appear above the long division symbol. We begin the division by asking What monomial, when multiplied by, gives 2 2? Because? 2 2 2, the answer is 2. We place 2 in the quotient, multiply each term of the divisor by 2, subtract, and bring down the q We continue the division by asking What monomial, when multiplied by, gives 23? We place the answer, 23, in the quotient, multiply each term of the divisor by 23, and subtract. This time, there is no number to bring down q Because the degree of the remainder, 29, is less than the degree of the divisor, the division process stops, and we can epress the result in the form quotient remainder divisor

49 Section A.4 Polynomials A49 Thus, EXAMPLE 9 Using Long Division to Divide Polynomials Divide by 2 2. Comment solution In Eample 9, we could write the missing powers of using coefficients of q We set up the division, leaving spaces for the missing powers of in the dividend. 2 2q6 3 2 The division process continues as usual, with the following results: q Thus, self Check 9 Divide: 3 q EXAMPLE 0 solution now Try Eercise 3. Using Long Division to Divide Polynomials Divide by The division process works best when the terms in the divisor and dividend are written with their eponents in descending order q Thus, self Check 0 Divide: 2 q Not Now For Try Eercise. Sale

50 Not For Sale A0 Appendi A A Review of Basic Algebra Self Check Answers b 2 2 7a 2 b 4. a b c " " "4 h 2 Eercises A.4 Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice eercises. Fill in the blanks.. A is a real number or the product of a real number and one or more. 2. The of a monomial is the sum of the eponents of its. 3. A is a polynomial with three terms. 4. A is a polynomial with two terms.. A monomial is a polynomial with term. 6. The constant 0 is called the polynomial. 7. Terms with the same variables with the same eponents are called terms. 8. The of a polynomial is the same as the degree of its term of highest degree. 9. To combine like terms, we add their and keep the same and the same eponents. 0. The conjugate of 3" 2 is. Determine whether the given epression is a polynomial. If so, tell whether it is a monomial, a binomial, or a trinomial, and give its degree y y / y " y 3 7. " y 3 2 4y 2 2y 2 Practice Perform the operations and simplify y 2y y 2 2y t 7 2 7t t 7 2 3t t 2 2 2t 2 4t 2 2 3t t yy y 2 y 22 2 y2y a 2 a 2 3aa a 2 a y 2 4y2 2 y 2 3y2 y2 3y mnm 2n2 2 6m3mn 2 2 2n4mn y 3 4y a 3 b22a 2 b m 2 n2mn 2 2 a2 mn 2 b r2 s 3 a2r2 s 3 b ars2 2 b rsr 2 s u 2 v2uv 2 2 y ab 2 c2ac 3bc 2 2 4ab 2 c mn2 2 4mn 2 6m a 22 a y 2 2 y a t z 72 z 2 72

51 Section A.4 Polynomials A z 42 z u 22 3u "2 2 "3 2 "3 92. "3 2 "2 " a 2 2b a b2 4a 2 b2 93. " 2 "y " "y 94. "2 y "2 2 y. 3m 4n2 3m 2 4n2 6. 4r 3s y y y2 3 y y y a 2 b2 a 2b2 6. z 2t2 z 2 2 t2 62. y y2 63. Q" 3RQ2 2 "R 64. Q"2 RQ3 "2R y y 4y r 2 3s2 2r 2 4rs 2 2s 2 2 Multiply the epressions as you would multiply polynomials. 7. 2y n 3y n y 2n a 2n 2a n 3a n2 2 Rationalize each numerator. 9. " y 2 "3 y " n y n 2 2n y 2n 3 22n y n a 3n b 2n a 23n b 2 ab 22n 2 7. n 32 n y y 2y 76. a n 2 2 a n 60 y r n r n a3 b 4 27a 2 b 4 2 8a 2 b 3 8a 2 b z n 32 3z n /2 /2 y y / ab /2 a /2 b /2 b / a /2 b /2 2 a /2 2 b / q /2 y / q Rationalize each denominator q "3 2 " y "7 2 " y q "3 2y "7 89. y " " q Not For Sale y 2 "2 " " 3 2 " " "a 2 "b "a "b "2 h 2 "2 h Perform each division and write all answers without using negative eponents a2 b 3 8ab r2 s t 3 27r 6 s 2 t y 4 z y 6 z m6 n 4 p 2 26m 6 n 7 p y 2 3 y y m4 n 9 2 6m 3 n 4 2m 3 n 3 Perform each division. If there is a nonzero remainder, write the answer in quotient remainder divisor form q3 2 6

52 Not For Sale A2 Appendi A A Review of Basic Algebra q q applications 23. Geometry Find an epression that represents the area of the brick wall. ( 2) ft ( + ) ft 24. Geometry The area of the triangle shown in the illustration is represented as square feet. Find an epression that represents its height. Height ( + 8) ft 2. Gift Boes The corners of a 2 in.-by-2 in. piece of cardboard are folded inward and glued to make a bo. Write a polynomial that represents the volume of the resulting bo. Crease here and fold inward 26. Travel Complete the following table, which shows the rate (mph), time traveled (hr), and distance traveled (mi) by a family on vacation. r? t d discovery and Writing 27. Show that a trinomial can be squared by using the formula a b c2 2 a 2 b 2 c 2 2ab 2bc 2ac. 28. Show that a b c d2 2 a 2 b 2 c 2 d 2 2ab 2ac 2ad 2bc 2bd 2cd. 29. Eplain the FOIL method. 30. Eplain how to rationalize the numerator of " Eplain why a b2 2 2 a 2 b Eplain why "a 2 b 2 2 "a 2 "b 2. Review Simplify each epression. Assume that all variables represent positive numbers /2 34. a 8 22/3 2 b 3. a 624 3/4 6y 8 b 36. " " 3 6ab 4 2 b" 3 4ab 38. " 4,280 " in. 2 in.

53 Section A. Factoring Polynomials A3 A. Factoring Polynomials EXAMPLE Peter Coombs/Alamy In this section, we will learn to. Factor out a common monomial. 2. Factor by grouping. 3. Factor the difference of two squares. 4. Factor trinomials.. Factor trinomials by grouping. 6. Factor the sum and difference of two cubes. 7. Factor miscellaneous polynomials. The television network series CSI: Crime Scene Investigation and its spinoffs, CSI: NY and CSI: Miami, are favorite television shows of many people. Their fans enjoy watching a team of forensic scientists uncover the circumstances that led to an unusual death or crime. These mysteries are captivating because many viewers are interested in the field of forensic science. In this section, we will investigate a mathematics mystery. We will be given a polynomial and asked to unveil or uncover the two or more polynomials that were multiplied together to obtain the given polynomial. The process that we will use to solve this mystery is called factoring. When two or more polynomials are multiplied together, each one is called a factor of the resulting product. For eample, the factors of are 7, 2, and 3 The process of writing a polynomial as the product of several factors is called factoring. In this section, we will discuss factoring where the coefficients of the polynomial and the polynomial factors are integers. If a polynomial cannot be factored by using integers only, we call it a prime polynomial.. Factor Out a Common Monomial The simplest type of factoring occurs when we see a common monomial factor in each term of the polynomial. In this case, our strategy is to use the Distributive Property and factor out the greatest common factor. Factoring by Removing a Common Monomial Factor: 3y 2 6. solution We note that each term contains a greatest common factor of 3: 3y 2 6 3y We can then use the Distributive Property to factor out the common factor of 3: 3y 2 6 3y 2 22 We can check by multiplying: 3y y 2 6. self Check Factor: 4a 2 2 8ab. Now Try Eercise. Not For Sale

54 Not For Sale A4 Appendi A A Review of Basic Algebra EXAMPLE 2 solution Factoring by Removing a Common Monomial Factor: 2 y 2 z 2 2 yz. We factor out the greatest common factor of yz: 2 y 2 z 2 2 yz yzyz2 2 yz2 yzyz 2 2 We can check by multiplying. The last term in the epression 2 y 2 z 2 2 yz has an understood coefficient of. When the yz is factored out, the must be written. self Check 2 Factor: a 2 b 2 c 2 a 3 b 3 c 3. EXAMPLE 3 solution now Try Eercise Factor by Grouping A strategy that is often helpful when factoring a polynomial with four or more terms is called grouping. Terms with common factors are grouped together and then their greatest common factors are factored out using the Distributive Property. Factoring by Grouping Factor: a b a b. Although there is no factor common to all four terms, we can factor out of the first two terms and write the epression as a b a b a b2 a b2 We can now factor out the common factor of a b. a b a b a b2 a b2 self Check 3 Factor: 2 y 2 2y. Now Try Eercise 7. a b2 a b2 a b2 2 We can check by multiplying. 3. Factor the Difference of Two Squares A binomial that is the difference of the squares of two quantities factors easily. The strategy used is to write the polynomial as the product of two factors. The first factor is the sum of the quantities and the other factor is the difference of the quantities. EXAMPLE 4 solution Factoring the Difference of Two Squares Factor: We observe that each term is a perfect square:

55 Section A. Factoring Polynomials A The difference of the squares of two quantities is the product of two factors. One is the sum of the quantities, and the other is the difference of the quantities. Thus, factors as We can check by multiplying. self Check 4 Factor: 9a 2 2 6b 2. Now Try Eercise 2. Caution Factoring the Difference of Two Squares example solution If you are limited to integer coefficients, the sum of two squares cannot be factored. For eample, 2 y 2 is a prime polynomial. Eample 4 suggests a formula for factoring the difference of two squares. 2 2 y 2 y2 2 y2 Factoring the Difference of Two Squares Twice in One Problem Factor: 6m 4 2 n 4. The binomial 6m 4 2 n 4 can be factored as the difference of two squares: 6m 4 2 n 4 4m n m 2 n 2 2 4m 2 2 n 2 2 The first factor is the sum of two squares and is prime. The second factor is a difference of two squares and can be factored: self Check Factor: a 4 2 8b 4. EXAMPLE 6 6m 4 2 n 4 4m 2 n m2 2 2 n 2 4 now Try Eercise 23. 4m 2 n 2 2 2m n2 2m 2 n2 We can check by multiplying. Removing a Common Factor and Factoring the Difference of Two Squares Factor: 8t solution We begin by factoring out the common monomial factor of 2. 8t t Since 9t is the difference of two squares, it can be factored. 8t t t 42 3t 2 42 We can check by multiplying. self Check 6 Factor: Not Now For Try Eercise 6. Sale

56 Not For Sale A6 Appendi A A Review of Basic Algebra 4. Factor Trinomials Trinomials that are squares of binomials can be factored by using the following formulas. Factoring Trinomial Squares () 2 2y y 2 y2 y2 y2 2 (2) 2 2 2y y 2 2 y2 2 y2 2 y2 2 For eample, to factor a 2 2 6a 9, we note that it can be written in the form a a2 3 2 a and y 3 which matches the left side of Equation 2 above. Thus, a 2 2 6a 9 a a2 3 2 a 2 32 a 2 32 a We can check by multiplying. Factoring trinomials that are not squares of binomials often requires some guesswork. If a trinomial with no common factors is factorable, it will factor into the product of two binomials. EXAMPLE 7 Factoring a Trinomial with Leading Coefficient of solution Factor: To factor , we must find two binomials a and b such that a2 b2 where the product of a and b is 0 and the sum of a and b is 3. ab 20 and a b 3 To find such numbers, we list the possible factorizations of 0: Only in the factorization 222 do the factors have a sum of 3. Thus, a and b 22, and a2 b2 (3) We can check by multiplying. Because of the Commutative Property of Multiplication, the order of the factors in Equation 3 is not important. Equation 3 can also be written as self Check 7 Factor: p 2 2 p 2 6. Now Try Eercise 29. EXAMPLE 8 Factoring a Trinomial with Leading Coefficient not Factor:

57 Section A. Factoring Polynomials A7 solution Since the first term is 2 2, the first terms of the binomial factors must be 2 and : The product of the last terms must be 26, and the sum of the products of the outer terms and the inner terms must be 2. Since the only factorization of 26 that will cause this to happen is 3222, we have We can check by multiplying. self Check 8 Factor: Now Try Eercise 33. strategy for Factoring a General Trinomial with Integer Coefficients EXAMPLE 9 solution It is not easy to give specific rules for factoring trinomials, because some guesswork is often necessary. However, the following hints are helpful.. Write the trinomial in descending powers of one variable. 2. Factor out any greatest common factor, including 2 if that is necessary to make the coefficient of the first term positive. 3. When the sign of the first term of a trinomial is and the sign of the third term is, the sign between the terms of each binomial factor is the same as the sign of the middle term of the trinomial. When the sign of the first term is and the sign of the third term is 2, one of the signs between the terms of the binomial factors is and the other is Try various combinations of first terms and last terms until you find one that works. If no possibilities work, the trinomial is prime.. Check the factorization by multiplication. Factoring a Trinomial Completely Factor: 0y 24y We write the trinomial in descending powers of and then factor out the common factor of 22. 0y 24y y 24y y 2 2y 2 2 Since the sign of the third term of y 2 2y 2 is 2, the signs between the binomial factors will be opposite. Since the first term is 3 2, the first terms of the binomial factors must be 3 and : y 2 2y The product of the last terms must be 22y 2, and the sum of the outer terms and the inner terms must be 2y. Of the many factorizations of 22y 2, only 4y23y2 leads to a middle term of 2y. So we have 0y 24y y 24y y 2 2y y2 2 3y2 Not For Sale We can check by multiplying.

58 Not For Sale A8 Appendi A A Review of Basic Algebra self Check 9 Factor: y 2 6y 2. Now Try Eercise 69. EXAMPLE 0. Factor Trinomials by Grouping Another way of factoring trinomials involves factoring by grouping. This method can be used to factor trinomials of the form a 2 b c. For eample, to factor , we note that a 6, b, and c 26, and proceed as follows:. Find the product ac: This number is called the key number. 2. Find two factors of the key number 2362 whose sum is b. Two such numbers are 9 and and Use the factors 9 and 24 as coefficients of two terms to be placed between 6 2 and Factor by grouping: Factoring a Trinomial by Grouping Factor: Factor out 2 3. solution Since a and c 22 in the trinomial, ac 230. We now find factors of 230 whose sum is b. Such factors are 6 and 2. We use these factors as coefficients of two terms to be placed between 2 and Finally, we factor by grouping. self Check 0 Factor: a 2 7a Now Try Eercise 39. We can often factor polynomials with variable eponents. For eample, if n is a natural number, because a 2n 2 a n 2 6 a n 2 a n 2 62 a n 2 a n 2 62 a 2n 2 6a n a n 2 6 a 2n 2 a n 2 6 Combine like terms. 6. Factor the Sum and Difference of Two Cubes Two other types of factoring involve binomials that are the sum or the difference of two cubes. Like the difference of two squares, they can be factored by using a formula.

59 Section A. Factoring Polynomials A9 Factoring the Sum and Difference of Two Cubes 3 y 3 y2 2 2 y y y 3 2 y2 2 y y 2 2 example solution Factoring the Sum of Two Cubes Factor: y 3. We can write this epression as the sum of two cubes and factor it as follows: y y2 3 self Check Factor: 8a 3,000b 6. now Try Eercise 4. example 2 solution 3 2 4y y2 4y y y 6y 2 2 We can check by multiplying. Factoring the Difference of Two Cubes Factor: This binomial can be written as , which is the difference of two cubes. Substituting into the formula for the difference of two cubes gives self Check 2 Factor: p Now Try Eercise 43. EXAMPLE 3 solution We can check by multiplying. 7. Factor Miscellaneous Polynomials Factoring a Miscellaneous Polynomial Factor: 2 2 y Here we will factor a trinomial and a difference of two squares. 2 2 y y 2 Use the Commutative Property to rearrange the terms y 2 Factor y2 3 2 y2 Factor the difference of two squares. We could try to factor this epression in another way. 2 2 y y2 2 y Factor 2 2 y 2 and 6 9. However, we are unable to finish the factorization. If grouping in one way doesn t work, try various other ways. self Check 3 Factor: a 2 8a 2 b 2 6. Not For Sale Now Try Eercise 97.

60 Not For Sale A60 Appendi A A Review of Basic Algebra EXAMPLE 4 solution Factoring Strategy Factoring a Miscellaneous Trinomial Factor: z 4 2 3z 2. This trinomial cannot be factored as the product of two binomials, because no combination will give a middle term of 23z 2. However, if the middle term were 22z 2, the trinomial would be a perfect square, and the factorization would be easy: z 4 2 2z 2 z z z We can change the middle term in z 4 2 3z 2 to 22z 2 by adding z 2 to it. However, to make sure that adding z 2 does not change the value of the trinomial, we must also subtract z 2. We can then proceed as follows. z 4 2 3z 2 z 4 2 3z 2 z 2 2 z 2 Add and subtract z 2. z 4 2 2z 2 2 z 2 Combine 23z 2 and z 2. z z 2 Factor z 4 2 2z 2. z 2 2 z2 z z2 Factor the difference of two squares. In this type of problem, we will always try to add and subtract a perfect square in hopes of making a perfect-square trinomial that will lead to factoring a difference of two squares. self Check 4 Factor: Now Try Eercise 03. It is helpful to identify the problem type when we must factor polynomials that are given in random order.. Factor out all common monomial factors. 2. If an epression has two terms, check whether the problem type is a. The difference of two squares: 2 2 y 2 y2 2 y2 b. The sum of two cubes: 3 y 3 y2 2 2 y y 2 2 c. The difference of two cubes: 3 2 y 3 2 y2 2 y y If an epression has three terms, try to factor it as a trinomial. 4. If an epression has four or more terms, try factoring by grouping.. Continue until each individual factor is prime. 6. Check the results by multiplying. Self Check Answers. 4aa 2 2b2 2. a 2 b 2 c 2 abc2 3. y a 4b2 3a 2 4b2. a 2 9b 2 2 a 3b2 a 2 3b p 2 62 p y2 2 y2 0. 3a 42 a a b 2 2 a 2 2 ab 2 2b p 2 42 p 2 4p a 4 b2 a 4 2 b

61 Section A. Factoring Polynomials A6 Eercises A. Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice eercises. Fill in the blanks.. When polynomials are multiplied together, each polynomial is a of the product. 2. If a polynomial cannot be factored using coefficients, it is called a polynomial. Complete each factoring formula. 3. a b y y y y y y y 3 Practice In each epression, factor out the greatest common monomial y y 3 6y y y y 2 z 2 yz 2 In each epression, factor by grouping.. a y2 b y2 6. b 2 y2 a 2 y2 7. 4a b 2 2a 2 2 3ab y 4y In each epression, factor the difference of two squares z r z y y 2 6y y 2 3y 2 In each epression, factor the trinomial by grouping p 2 7p q 2 2 9q 2 In each epression, factor the sum of two cubes. 4. t r 3 8s 3 In each epression, factor the difference of two cubes z a Factor each epression completely. If an epression is prime, so indicate. 4. 3a 2 bc 6ab 2 c 9abc y 3 z y 2 z 2 2 2yz y 2 9y ty 2ct 2 3ty 2 3ct 0. 2a 4ay 2 b 2 2by. a b ay by az bz y 3 8y 3 2 y y 2 z z 2 2 y y2 2 2 y a a y 4 8. z y 2 3y 6. 8y a 24a y 3y y 6y 2 In each epression, factor the trinomial p 2 2 8pq 2 70q y 2 9y a 2 2 2a m 2 47mn 2 3n r 2 2 rs s b 2 2 0b y 2 4y y 3 2 y 2 90y 3. m 2 4mn Not 4n r 2 2 8rs For 6s 2 Sale

62 Not For Sale A62 Appendi A A Review of Basic Algebra a 2n 2 2a n a 2n 6a n 8 0. Movie Stunts The formula that gives the distance a stuntwoman is above the ground t seconds after she falls over the side of a 44-foot tall building is f t 2. Factor the right side n 2 7 n n 9 n n 2 9y 2n n 2 2 n y 2n 2 y n y 4n 2 2y 2n , y y y a 6 2 y a 6 b a 3 2 b 3 a 2 b 94. a 2 2 y a y y z 2 6z y y y a b a b a b2 2 2 a b y 4 2y a 4 3a Candy To find the amount of chocolate used in the outer coating of one of the malted-milk balls shown, we can find the volume V of the chocolate shell using the formula V 4 3 pr pr 2 3. Factor the epression on the right side of the formula. Inner radius r 2 Outer radius r Istockphoto.com/maureenpr 44 ft discovery and Writing. Eplain how to factor the difference of two squares. 2. Eplain how to factor the difference of two cubes. 3. Eplain how to factor a 2 2 b 2 a b. 4. Eplain how to factor 2 2. Factor the indicated monomial from the given epression.. 3 2; ; ; ; 3 9. a b; a 20. a 2 b; b 2. /2 ; / /2 2 /2 ; / "2y; "2 24. "3a 2 3b; "3 2. ab 3/2 2 a 3/2 b; ab 26. ab 2 b; b 2 Factor each epression by grouping three terms and two terms y 2 2y y 2 2y 29. a 4 2a 3 a 2 a 30. a 4 a 3 2 2a 2 a 2 review 3. Which natural number is neither prime nor composite? 32. Graph the interval 322, Simplify: Simplify: a3 2 3 a a 2 a a b 36. Simplify: " Simplify: "20 2 " Rationalize the denominator: 3! 3 3.

63 Section A.6 Rational Epressions A63 A.6 Rational Epressions In this section, we will learn to. Define rational epressions. 2. Simplify rational epressions. 3. Multiple and divide rational epressions. 4. Add and subtract rational epressions.. Simplify comple fractions. Abercrombie and Fitch (A&F) is a very successful American clothing company founded in 892 by David Abercrombie and Ezra Fitch. Today the stores are popular shopping destinations for university students wanting to keep up with the latest styles and trends. Suppose that a clothing manufacturer finds that the cost in dollars of producing fleece vintage shirts is given by the algebraic epression 3,000. The average cost of producing each shirt could be obtained by dividing the production cost, 3,000, by the number of shirts produced,. The algebraic fraction 3,000 represents the average cost per shirt. We see that the average cost of producing 200 shirts would be $ , An understanding of algebraic fractions is important in solving many real-life problems. Photo by Tim Boyle/Getty Images Caution Remember that the denominator of a fraction cannot be zero.. Define Rational Epressions If and y are real numbers, the quotient y y 2 02 is called a fraction. The number is called the numerator, and the number y is called the denominator. Algebraic fractions are quotients of algebraic epressions. If the epressions are polynomials, the fraction is called a rational epression. The first two of the following algebraic fractions are rational epressions. The third is not, because the numerator and denominator are not polynomials. y 2 2y 8ab 2 2 6c 3 /2 4 Not For Sale y 2 2 3y /2 2 /2