Slide 1- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Foundations of Algebra CHAPTER 1.1Number Sets and the Structure of Algebra 1.2Fractions 1.3Adding and Subtracting Real Numbers; Properties of Real Numbers 1.4Multiplying and Dividing Real Numbers; Properties of Real Numbers 1.5Exponents, Roots, and Order of Operations 1.6Translating Word Phrases to Expressions 1.7Evaluating and Rewriting Expressions 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Number Sets and the Structure of Algebra Understand the structure of algebra. 2.Classify number sets. 3.Graph rational numbers on a number line. 4.Determine the absolute value of a number. 5.Compare numbers.

Slide 1- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Understand the structure of algebra.

Slide 1- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions Variable: A symbol that can vary in value. Constant: A symbol that does not vary in value. Variables are usually letters of the alphabet, like x or y. Usually constants are symbols for numbers, like 1, 2, ¾, 6.74, and so on.

Slide 1- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Expression: A constant, variable, or any combination of constants, variables, and arithmetic operations that describe a calculation. Examples of expressions: or 4x  5 or

Slide 1- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Equation: A mathematical relationship that contains an equal sign. Examples of equations: = 8 or 4x  5 = 12 or

Slide 1- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inequality: A mathematical relationship that contains an inequality symbol ( ,, , or  ). Symbolic formTranslation 8  3 Eight is not equal to three. 5 < 7Five is less than seven. 7 > 5Seven is greater than five. x  3 x is less than or equal to three. y  2 y is greater than or equal to two.

Slide 1- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Classify number sets.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Set: A collection of objects. Braces are used to indicate a set. For example, the set containing the numbers 1, 2, 3, and 4 would be written {1, 2, 3, 4}. The numbers 1, 2, 3, and 4 are called the members or elements of this set.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Writing Sets To write a set, write the members or elements of the set separated by commas within braces, { }.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write the set containing the first four days of the week. Answer: {Sunday, Monday, Tuesday, Wednesday}

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Numbers are classified using number sets. Natural numbers contain the counting numbers 1, 2, 3, 4, …and is written {1, 2, 3, …}. The three dots are called ellipses and indicate that the numbers continue forever in the same pattern. Whole numbers: natural numbers and 0 {0, 1, 2, 3,…} Integers: whole numbers and the opposite (or negative) of every natural number {…,  3,  2,  1, 0, 1, 2, 3…} Rational: every real number that can be expressed as a ratio of integers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational number: Any real number that can be expressed in the form, where a and b are integers and b  0.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Determine whether the given number is a rational number. a. b. 0.8c. Answer: a. Yes, because 5 and 6 are integers. b. 0.8 Yes, 0.8 can be expressed as a fraction 8 over 10, and 8 and 10 are integers. c. The bar indicates that the digit repeats. This is the decimal equivalent of 1 over 3. Yes this is a rational number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Irrational number: Any real number that is not rational. Examples:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Graph rational numbers on a number line.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph on a number line. Answer The number is located 4/5 of the way between 2 and The number line is divided into 5 equally spaced divisions. Place a dot on the 4 th mark.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Determine the absolute value of a number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute value: A given number’s distance from 0 on a number line. The absolute value of a number n is written |n|. The absolute value of 5 is 5.of  5 is 5. |5| = 5|  5| = 5  5 units from 0 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Value The absolute value of every real number is either positive or 0.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. a. |  9.4|b. Answer a. |  9.4| = 9.4 b.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 5 Compare numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Comparing Numbers For any two real numbers a and b, a is greater than b if a is to the right of b on a number line. Equivalently, b is less than a if b is to the left of a on a number line.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use =, to write a true statement. a. 3 ___  3b.  1.8 ___  1.6 Answer a. 3 ___  3 3 >  3 because 3 is farther right on a number line. b.  1.8 ___  1.6  1.8 <  1.6 because –1.8 is further to the left on a number line.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To which set of numbers does  6 belong? a) Irrational b) Natural and whole numbers c) Natural numbers, whole numbers, and integers d) Integers and rational numbers

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To which set of numbers does  6 belong? a) Irrational b) Natural and whole numbers c) Natural numbers, whole numbers, and integers d) Integers and rational numbers

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify |7|. a) 7 b)  7 c) 0 d) 1/7

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify |7|. a) 7 b)  7 c) 0 d) 1/7

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Which statement is false? a) 7 > 4 b)  2.4 >  1.4 c) 10 < 22 d)  3.6 >  6.4

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Which statement is false? a) 7 > 4 b)  2.4 >  1.4 c) 10 < 22 d)  3.6 >  6.4

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fractions Write equivalent fractions. 2.Write equivalent fractions with the LCD. 3.Write the prime factorization of a number. 4.Simplify a fraction to lowest terms.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fraction: A quotient of two numbers or expressions a and b having the form where b  0. The top number in a fraction is called the numerator. The bottom number is called the denominator. Fractions indicated part of a whole.  Numerator  Denominator

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Write equivalent fractions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Writing Equivalent Fractions For any fraction, we can write an equivalent fraction by multiplying or dividing both its numerator and denominator by the same nonzero number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the missing number that makes the fractions equivalent. a.b. Solution a.b. Multiply the numerator and denominator by 3. Divide the numerator and denominator by 6.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Write equivalent fractions with the LCD.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiple: A multiple of a given integer n is the product of n and an integer. We can generate multiples of a given number by multiplying the given number by the integers. Multiples of 2Multiples of 3

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Least common multiple (LCM): The smallest number that is a multiple of each number in a given set of numbers. Least common denominator (LCD): The least common multiple of the denominators of a given set of fractions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write as equivalent fractions with the LCD. Solution The LCD of 8 and 6 is 24.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Write the prime factorization of a number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factors: If a  b = c, then a and b are factors of c. Example: 6  7 = 42, 6 and 7 are factors of 42 Prime number: A natural number that has exactly two different factors, 1 and the number itself. Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,… Prime factorization: A factorization that contains only prime factors.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the prime factorization of 420. Solution Factor 420 to 10 and 42. (Any two factors will work.) Factor 10 to 2 and 5, which are primes. Then factor 42 to 6 and 7. 7 is prime and then factor 6 into 2 and 3, which are primes. Answer 2  2  3  5  7

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Simplify a fraction to lowest terms.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Lowest terms: Given a fraction and b  0, if the only factor common to both a and b is 1, then the fraction is in lowest terms.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplifying a Fraction with the Same Nonzero Numerator and Denominator Eliminating a Common Factor in a Fraction

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley These rules allow us to write fractions in lowest terms using prime factorization. The idea is to replace the numerator and denominator with their prime factorizations and then eliminate the prime factors that are common to both the numerator and denominator.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplifying a Fraction to Lowest Terms To simplify a fraction to lowest terms: 1. Replace the numerator and denominator with their prime factorizations. 2. Eliminate (divide out) all prime factors common to the numerator and denominator. 3. Multiply the remaining factors.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify to lowest terms. Solution Replace the numerator and denominator with their prime factorizations, then eliminate the common prime factors.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example At a company, 225 of the 1050 employees have optional eye insurance coverage as part of their benefits package. What fraction of the employees have optional eye insurance coverage? Solution Answer 3 out of 14 employees have optional eye insurance.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the prime factorization of 360? a) 6  6  5 b) 2 3  3 2  5 c) 2 2  3 2  5 d) 3 2  5  7

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the prime factorization of 360? a) 6  6  5 b) 2 3  3 2  5 c) 2 2  3 2  5 d) 3 2  5  7

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify to lowest terms: a) b) c) d)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify to lowest terms: a) b) c) d)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding and Subtracting Real Numbers; Properties of Real Numbers Add integers. 2.Add rational numbers. 3.Find the additive inverse of a number. 4.Subtract rational numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Add integers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parts of an addition statement: The numbers added are called addends and the answer is called a sum = 5 addendssum

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Properties of Addition Symbolic FormWord Form Additive Identity a + 0 = aThe sum of a number and 0 is that number. Commutative Property of Addition a + b = b + aChanging the order of addends does not affect the sum. Associative Property of Addition a + (b + c) = (a + b) + cChanging the grouping of three or more addends does not affect the sum.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding Numbers with the Same Sign To add two numbers that have the same sign, add their absolute values and keep the same sign.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. a b. –16 + (– 22) Solution a = 39 b. –16 + (–22) = –38

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding Numbers with Different Signs To add two numbers that have different signs, subtract their absolute values and keep the sign of the number with the greater absolute value.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. a (–17)b. – Solution a (–17) = 18 b. – = –22

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Add rational numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding Fractions with the Same Denominator To add fractions with the same denominator, add the numerators and keep the same denominator, then simplify.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Add. a. b. Solution a. b. Replace 6 and 9 with their prime factorizations, divide out the common factor, 3, then multiply the remaining factors. Simplify to lowest terms by dividing out the common factor, 2.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding Fractions To add fractions with different denominators: 1.Write each fraction as an equivalent fraction with the LCD. 2.Add the numerators and keep the LCD. 3.Simplify.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Write equivalent fractions with 24 in the denominator. Add numerators and keep the common denominator. Because the addends have the same sign, we add and keep the same sign.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Find the additive inverse of a number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Additive inverse: Two numbers whose sum is zero. What happens if we add two numbers that have the same absolute value but different signs, such as 5 + (–5)? In money terms, this is like making a $5 payment towards a debt of $5. Notice the payment pays off the debt so that the balance is (–5) = 0 Because their sum is zero, we say 5 and –5 are additive inverses, or opposites.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the additive inverse of the given number. a. 8b. –2c. 0 Answers a. –8because 8 + (–8) = 0 b. 2 because – = 0 c. 0 because = 0

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Subtract rational numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parts of a subtraction statement: 8 – 5 = 3 Minuend Subtrahend Difference

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rewriting Subtraction To write a subtraction statement as an equivalent addition statement, change the operation symbol from a minus sign to a plus sign, and change the subtrahend to its additive inverse.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Subtract a. –17 – (–5) Solution Write the subtraction as an equivalent addition. –17 – (–5) = – = –12 Change the operation from minus to plus. Change the subtrahend to its additive inverse.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example b – 9.03 Solution Write the equivalent addition statement – 9.03 = (– 9.03) = –4.96

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add –6 + (–9). a) –15 b)  3 c) 3 d) 15

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add –6 + (–9). a) –15 b)  3 c) 3 d) 15

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract 5 – (–8). a) –13 b)  3 c) 3 d) 13

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract 5 – (–8). a) –13 b)  3 c) 3 d) 13

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract a) b) c) d)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract a) b) c) d)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying and Dividing Real Numbers; Properties of Real Numbers Multiply integers. 2.Multiply more than two numbers. 3.Multiply rational numbers. 4.Find the multiplicative inverse of a number. 5.Divide rational numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Multiply integers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In a multiplication statement, factors are multiplied to equal a product. Product Factors

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Properties of Multiplication Symbolic FormWord Form Multiplicative Property of 0 The product of a number multiplied by 0 is 0. Multiplicative Identity The product of a number multiplied by 1 is the number. Commutative Property of Multiplication ab=baChanging the order of factors does not affect the product. Associative Property of Multiplication a(bc) = (ab)cChanging the grouping of three or more factors does not affect the product. Distributive Property of Multiplication over Addition a(b + c) =ab + acA sum multiplied by a factor is equal to the sum of that factor multiplied by each addend.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Two Numbers with Different Signs When multiplying two numbers that have different signs, the product is negative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. 7(–4)b. (–15)3 Solution a. 7(–4) = b. (–15)3 = Warning: Make sure you see the difference between 7(–4), which indicates multiplication, and 7 – 4, which indicates subtraction. –28 –45

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Two Numbers with the Same Sign When multiplying two numbers that have the same sign, the product is positive.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. –5(–9)b. (–6)(–8) Solution a. –5(–9) = b. (–6)(–8) = 45 48

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Multiply more than two numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying with Negative Factors The product of an even number of negative factors is positive, whereas the product of an odd number of negative factors is negative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. (–1)(–3)(–6)(7) Solution Because there are three negative factors (an odd number of negative factors), the result is negative. (–1)(–3)(–6)(7) = –126 b. (–2)(–4)(2)(–5)(–3) Solution Because there are four negative factors(an even number of negative factors), the result is positive. (–2)(–4)(2)(–5)(–3) = 240

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Multiply rational numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Fractions

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply Solution Divide out the common factor, 3. Because we are multiplying two numbers that have different signs, the product is negative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying Decimal Numbers To multiply decimal numbers: 1. Multiply as if they named whole numbers. 2.Place the decimal in the product so that it has the same number of decimal places as the total number of decimal places in the factors.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply (–7.6)(0.04). Solution First, calculate the value and disregard signs for now places place Answer –0.304 When we multiply two numbers with different signs, the product is negative. 3 places

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Find the multiplicative inverse of a number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplicative inverse: Two numbers whose product is 1. and are multiplicative inverses because their product is 1. Notice that to write a number’s multiplicative inverse, we simply invert the numerator and denominator. Multiplicative inverses are also known as reciprocals.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the multiplicative inverse. a. Answer The multiplicative inverse of is because b. – 9 Answer The multiplicative inverse of –9 is because

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 5 Divide rational numbers.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividend Divisor Quotient Parts of a division statement:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Signed Numbers When dividing two numbers that have the same sign, the quotient is positive. When dividing two numbers that have different signs, the quotient is negative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide. a. b. Solution a. b.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Division Involving 0

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Fractions

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide Solution Write an equivalent multiplication. Divide out the common factor, 5. Because we are dividing two numbers that have different signs, the result is negative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Decimal Numbers To divide decimal numbers, set up a long division and consider the divisor. Case 1: If the divisor is an integer, then divide as if the dividend were a whole number and place the decimal point in the quotient directly above its position in the dividend. Case 2: If the divisor is a decimal number, then 1. Move the decimal point in the divisor to the right enough places to make the divisor an integer. 2. Move the decimal point in the dividend the same number of places.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Decimal Numbers continued 3. Divide the divisor into the dividend as if both numbers were whole numbers. Be sure to align the digits in the quotient properly. 4. Write the decimal point in the quotient directly above its position in the dividend. In either of the two cases, continue the division process until you get a remainder of 0 or a repeating digit or block of digits in the quotient.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply (–6)(–3)(7). a) 126 b)  126 c) –63 d) 63

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply (–6)(–3)(7). a) 126 b)  126 c) – 63 d) 63

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Divide a) b) c) d)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Divide a) b) c) d)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponents, Roots, and Order of Operations Evaluate numbers in exponential form. 2.Evaluate square roots. 3.Use the order-of-operations agreement to simplify numerical expressions. 4.Find the mean of a set of data.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Evaluate numbers in exponential form.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sometimes problems involve repeatedly multiplying the same number. In such problems, we can use an exponent to indicate that a base number is repeatedly multiplied. Exponent: A symbol written to the upper right of a base number that indicates how many times to use the base as a factor. Base: The number that is repeatedly multiplied.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When we write a number with an exponent, we say the expression is in exponential form. The expression is in exponential form, where the base is 2 and the exponent is 4. To evaluate, write 2 as a factor 4 times, then multiply. Exponent Base Four 2s

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluating an Exponential Form To evaluate an exponential form raised to a natural number exponent, write the base as a factor the number of times indicated by the exponent, then multiply.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate. (–9) 2 Solution The exponent 2 indicates we have two factors of –9. Because we multiply two negative numbers, the result is positive. (–9) 2 = (–9)(–9) = 81

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate. Solution The exponent 3 means we must multiply the base by itself three times.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluating Exponential Forms with Negative Bases If the base of an exponential form is a negative number and the exponent is even, then the product is positive. If the base is a negative number and the exponent is odd, then the product is negative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Evaluate square roots.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Roots are inverses of exponents. More specifically, a square root is the inverse of a square, so a square root of a given number is a number that, when squared, equals the given number. Square Roots Every positive number has two square roots, a positive root and a negative root. Negative numbers have no real-number square roots.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The symbol, called the radical, is used to indicate finding only the positive (or principal) square root of a given number. The given number or expression inside the radical is called the radicand. Radicand Radical Principal Square Root

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Square Roots Involving the Radical Sign The radical symbol denotes only the positive (principal) square root.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate the square root. a. b. c. Solution a. b. c. not a real number

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Use the order-of-operations agreement to simplify numerical expressions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Order-of- Operations Agreement Perform operations in the following order: 1. Grouping symbols: parentheses ( ), brackets [ ], braces { }, absolute value | |, and radicals. 2. Exponents/Roots from left to right, in order as they occur. 3. Multiplication/Division from left to right, in order as they occur. 4. Addition/Subtraction from left to right, in order as they occur.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution Calculate within the innermost parenthesis. Evaluate the exponential form, brackets, and square root. Multiply 5(3). Add Subtract 24 – 7.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Square Root of a Product or Quotient If a square root contains multiplication or division, we can multiply or divide first, then find the square root of the result, or we can find the square roots of the individual numbers, then multiply or divide the square roots. Square Root of a Sum or Difference When a radical contains addition or subtraction, we must add or subtract first, then find the root of the sum or difference.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution Subtract within the radical. Evaluate the exponential form and root. Divide. Multiply. Subtract.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sometimes fraction lines are used as grouping symbols. When they are, we simplify the numerator and denominator separately, then divide the results.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Find the mean of a set of data.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding the Arithmetic Mean To find the arithmetic mean, or average, of n numbers, divide the sum of the numbers by n. Arithmetic mean =

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Bruce has the following test scores in his biology class: 92, 96, 81, 89, 95, 93. Find the average of his test scores. Solution Divide the sum of the 6 scores by 6.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify using order of operations. a)  18 b) 6 c) 30 d) 36

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify using order of operations. a)  18 b) 6 c) 30 d) 36

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify using order of operations. a) b) c) d) undefined

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify using order of operations. a) b) c) d) undefined

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translating Word Phrases to Expressions Translate word phrases to expressions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Translating word phrases to Expressions

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translating Basic Phrases AdditionTranslationSubtractionTranslation The sum of x and 3 x + 3The difference of x and 3 x – 3 h plus kh + kh minus kh – k 7 added to tt + 77 subtracted from tt – 7 3 more than a number n + 33 less than a number n – 3 y increased by 2y + 2y decreased by 2y – 2 Note: Since addition is a commutative operation, it does not matter in what order we write the translation. Note: Subtraction is not a commutative operation; therefore, the way we write the translation matters.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translating Basic Phrases MultiplicationTranslationDivisionTranslation The product of x and 3 3x3xThe quotient of x and 3 x  3 h times khkh divided by k h  kh  k Twice a number2n2nh divided into k k  hk  h Triple the number 3n3nThe ratio of a to b a  ba  b Two-thirds of a number Note: Like addition, multiplication is a commutative operation: it does not matter in what order we write the translation. Note: Division is like subtraction in that it is not a commutative operation; therefore, the way we write the translation matters.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translating Basic Phrases ExponentsTranslationRootsTranslation c squaredc2c2 The square root of x The square of bb2b2 k cubedk3k3 The cube of bb3b3 n to the fourth power n4n4 y raised to the fifth power y5y5

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The key words sum, difference, product, and quotient indicate the answer for their respective operations. sum of x and 3 x + 3 difference of x and 3 product of x and 3quotient of x and 3 x – 3 x  3 x  3

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Translate to an algebraic expression. a. five more than two times a number Translation: 5 + 2n or 2n + 5 b. seven less than the cube of a number Translation: n 3 – 7 c. the sum of h raised to the fourth power and twelve Translation: h

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translating Phrases Involving Parentheses Sometimes the word phrases imply an order of operations that would require us to use parentheses in the translation. These situations arise when the phrase indicates that a sum or difference is to be calculated before performing a higher-order operation such as multiplication, division, exponent, or root.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Translate to an algebraic expression. a. seven times the sum of a and b Translation: 7(a + b) b. the product of a and b divided by the sum of w 2 and 4 Translation: ab  (w 2 + 4) or

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translate the phrase to an algebraic expression. Twelve less than three times a number a) 3n + 12 b) 12 – 3n c) 3n – 12 d) 3n  12

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translate the phrase to an algebraic expression. Twelve less than three times a number a) 3n + 12 b) 12 – 3n c) 3n – 12 d) 3n  12

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translate the phrase to an algebraic expression. The difference of a and b decreased by the sum of w and z a) (a – b) – (w + z) b) a – b – (w + z) c) ab – (w + z) d) (b – a) – (w + z)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Translate the phrase to an algebraic expression. The difference of a and b decreased by the sum of w and z a) (a – b) – (w + z) b) a – b – (w + z) c) ab – (w + z) d) (b – a) – (w + z)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluating and Rewriting Expressions Evaluate an expression. 2.Determine all values that cause an expression to be undefined. 3.Rewrite an expression using the distributive property. 4.Rewrite an expression by combining like terms.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Evaluate an expression.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluating an Algebraic Expression To evaluate an algebraic expression: 1. Replace the variables with their corresponding given numbers. 2. Calculate the numerical expression using the order of operations.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate x 2 – 0.4xy + 9, when x = 7 and y = –2. Solution x 2 – 0.4xy + 9 (7) 2 – 0.4(7)(–2) + 9 = 49 – 0.4(7)(–2) + 9 = 49 – (–5.6) + 9 = = 63.6 Replace x with 7 and y with –2. Begin calculating by simplifying the exponential form. Multiply. Write the subtraction as an equivalent addition. Add from left to right.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Determine all values that cause an expression to be undefined.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When evaluating a division expression in which the divisor or denominator contains a variable or variables, we must be careful about what values replace the variable(s). We often need to know what values could replace the variable(s) and cause the expression to be undefined or indeterminate.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Determine all values that cause the expression to be undefined. a. b. Answer a. If x =  4, we have which is undefined because the denominator is 0. b. If x =  2 or 9 the denominator will be undefined.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Rewrite an expression using the distributive property.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Distributive Property of Multiplication over Addition a(b + c) = ab + ac This property gives us an alternative to the order of operations. 2(5 + 6) = 2(11)2(5 + 6) = 2   6 = 22 = = 22

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the distributive property to write an equivalent expression. a. 3(x + 3)b. –5(w – 4) Solution a. 3(x + 3) = 3  x + 3  3 = 3x + 9 b. –5(w – 4) = –5  w – (–5)  4 = –5w + 20

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Rewrite an expression by combining like terms.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Terms: Expressions that are the addends in an expression that is a sum. Coefficient: The numerical factor in a term. The coefficient of 5x 3 is 5. The coefficient of –8y is –8. Like terms: Variable terms that have the same variable(s) raised to the same exponents, or constant terms. Like termsUnlike terms 4x and 7x2x and 8y different variables 5y 2 and 10y 2 7t 3 and 3t 2 different exponents 8xy and 12xyx 2 y and xy 2 different exponents 7 and 1513 and 15x different variables

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Combining Like Terms To combine like terms, add or subtract the coefficients and keep the variables and their exponents the same.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Combine like terms. a. 10y + 8y Solution10y + 8y = 18y b. 8x – 3x Solution8x – 3x = 5x c. 13y 2 – y 2 Solution13y 2 – y 2 = 12y 2

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Combine like terms in 5y y 2 – 7. Solution 5y y 2 – 7 = 5y 2 + 4y – 7 Combine like terms. = 9y 2 – 1

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Combine like terms in 18y + 7x – y – 7x. Solution 18y + 7x – y – 7x = 17y + 0 = 17y

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Combine like terms in Solution Collect like terms. Write the fraction coefficients as equivalent fractions with their LCD, 12. Combine like terms.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate the expression 4(a + b) when a = 3 and b = –2. a) 4 b)  4 c) 12 d) 20

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate the expression 4(a + b) when a = 3 and b = –2. a) 4 b)  4 c) 12 d) 20

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For which values is the expression undefined? a) 8 b)  2 c)  2 and 5 d) 2 and  5

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For which values is the expression undefined? a) 8 b)  2 c)  2 and 5 d) 2 and  5

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify: 7x + 8 – 2x – 4 a) 9x – 4 b) 9x + 4 c) 5x – 4 d) 5x + 4

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify: 7x + 8 – 2x – 4 a) 9x – 4 b) 9x + 4 c) 5x – 4 d) 5x + 4