Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. 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- 3 Copyright © 2011 Pearson Education, Inc. Objective 1 Understand the structure of algebra.

Slide 1- 4 Copyright © 2011 Pearson Education, Inc. 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.

Slide 1- 5 Copyright © 2011 Pearson Education, Inc. 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- 6 Copyright © 2011 Pearson Education, Inc. Equation: A mathematical relationship that contains an equal sign. Examples of equations: = 8 or 4x  5 = 12 or

Slide 1- 7 Copyright © 2011 Pearson Education, Inc. 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- 8 Copyright © 2011 Pearson Education, Inc. Objective 2 Classify number sets.

Slide 1- 9 Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Writing Sets To write a set, write the members or elements of the set separated by commas within braces, { }.

Slide Copyright © 2011 Pearson Education, Inc. Example 1 Write the set containing the first four days of the week. Answer {Sunday, Monday, Tuesday, Wednesday}

Slide Copyright © 2011 Pearson Education, Inc. 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 ellipsis 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 © 2011 Pearson Education, Inc. Rational number: Any real number that can be expressed in the form, where a and b are integers and b  0.

Slide Copyright © 2011 Pearson Education, Inc. Example 2 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 © 2011 Pearson Education, Inc. Irrational number: Any real number that is not rational. Examples: Real numbers: The union of the rational and irrational numbers.

Slide Copyright © 2011 Pearson Education, Inc. Objective 3 Graph rational numbers on a number line.

Slide Copyright © 2011 Pearson Education, Inc. Example 3 Graph on a number line. Answer The number is located 4/5 of the way between 2 and Between 2 and 3, we divide the number line into 5 equally spaced divisions. Place a dot on the 4 th mark.

Slide Copyright © 2011 Pearson Education, Inc. Objective 4 Determine the absolute value of a number.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Absolute Value The absolute value of every real number is either positive or 0.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 Simplify. a. |  9.4|b. Answer a. |  9.4| = 9.4 b.

Slide Copyright © 2011 Pearson Education, Inc. Objective 5 Compare numbers.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 5 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 © 2011 Pearson Education, Inc. 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 1.1

Slide Copyright © 2011 Pearson Education, Inc. 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 1.1

Slide Copyright © 2011 Pearson Education, Inc. Simplify |7|. a) 7 b)  7 c) 0 d) 1/7 1.1

Slide Copyright © 2011 Pearson Education, Inc. Simplify |7|. a) 7 b)  7 c) 0 d) 1/7 1.1

Slide Copyright © 2011 Pearson Education, Inc. Which statement is false? a) 7 > 4 b)  2.4 >  1.4 c) 10 < 22 d)  3.6 > 

Slide Copyright © 2011 Pearson Education, Inc. Which statement is false? a) 7 > 4 b)  2.4 >  1.4 c) 10 < 22 d)  3.6 > 

Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Objective 1 Write equivalent fractions.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 1 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 © 2011 Pearson Education, Inc. Objective 2 Write equivalent fractions with the LCD.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 2 Write as equivalent fractions with the LCD. Solution The LCD of 8 and 6 is 24.

Slide Copyright © 2011 Pearson Education, Inc. Objective 3 Write the prime factorization of a number.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 3 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 © 2011 Pearson Education, Inc. Objective 4 Simplify a fraction to lowest terms.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Simplifying a Fraction with the Same Nonzero Numerator and Denominator Eliminating a Common Factor in a Fraction

Slide Copyright © 2011 Pearson Education, Inc. These rules allow us to write fractions in lowest terms using prime factorizations. 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 4a Simplify to lowest terms. Solution Replace the numerator and denominator with their prime factorizations; then eliminate the common prime factors.

Slide Copyright © 2011 Pearson Education, Inc. Example 4b Simplify to lowest terms. Solution Replace the numerator and denominator with their prime factorizations; then eliminate the common prime factors.

Slide Copyright © 2011 Pearson Education, Inc. Example 5 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 © 2011 Pearson Education, Inc. 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 1.2

Slide Copyright © 2011 Pearson Education, Inc. 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 1.2

Slide Copyright © 2011 Pearson Education, Inc. Simplify to lowest terms: a) b) c) d) 1.2

Slide Copyright © 2011 Pearson Education, Inc. Simplify to lowest terms: a) b) c) d) 1.2

Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Objective 1 Add integers.

Slide Copyright © 2011 Pearson Education, Inc. Parts of an addition statement: The numbers added are called addends and the answer is called a sum = 5 AddendsSum

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 1 Indicate whether each equation illustrates the additive identity, commutative property of addition, or the associative property of addition. a. (5 + 6) + 3 = 5 + (6 + 3) Answer Associative property of addition b. 0 + (  9) =  9 Answer Additive identity c. (  9 + 6) + 4 = 4 + (  9 + 6) Answer Commutative property of addition

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 2 Add. a b. –16 + (– 22) Solution a = 39 b. –16 + (–22) = –38

Slide Copyright © 2011 Pearson Education, Inc. Adding Numbers with Different Signs To add two numbers that have different signs, subtract the smaller absolute value from the greater absolute value and keep the sign of the number with the greater absolute value.

Slide Copyright © 2011 Pearson Education, Inc. Example 3 Add. a (–17)b. – Solution a (–17) = 18 b. – = –22

Slide Copyright © 2011 Pearson Education, Inc. Example 3 continued Add. c (–27)d. – Solution c (–27) = –12 d. – = –26

Slide Copyright © 2011 Pearson Education, Inc. Objective 2 Add rational numbers.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 4 Add. a. b. Solution a. 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, 3.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 continued Add. c. Solution a. Simplify to lowest terms by dividing out the common factor, 2.

Slide Copyright © 2011 Pearson Education, Inc. 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.

Solution Slide Copyright © 2011 Pearson Education, Inc. Example 5a Write equivalent fractions with 12 in the denominator. Add numerators and keep the common denominator. Because the addends have the same sign, we add and keep the same sign.

Solution Slide Copyright © 2011 Pearson Education, Inc. Example 5b Write equivalent fractions with 12 in the denominator. Add numerators and keep the common denominator. Because the addends have different signs, we subtract and keep the sign of the number with the greater absolute value.

Solution Slide Copyright © 2011 Pearson Education, Inc. Example 5c Write equivalent fractions with 120 in the denominator. Add numerators and keep the common denominator. Reduce to lowest terms.

Anna has a balance of $ and incurs a debt of $ What is Anna’s new balance? Solution A debt of $85.42 is  $ Her balance is (– 85.42) = $ Slide Copyright © 2011 Pearson Education, Inc. Example 6

Slide Copyright © 2011 Pearson Education, Inc. Objective 3 Find the additive inverse of a number.

Slide Copyright © 2011 Pearson Education, Inc. Additive inverses: Two numbers whose sum is 0. 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 © 2011 Pearson Education, Inc. Example 7 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 © 2011 Pearson Education, Inc. Example 8 Simplify. a. – (–5)b. –|2|c. –| –9| Answers a. – (–5) = 5 b. –|2| = –2 c. –| –9| = –9

Slide Copyright © 2011 Pearson Education, Inc. Objective 4 Subtract rational numbers.

Slide Copyright © 2011 Pearson Education, Inc. Parts of a subtraction statement: 8 – 5 = 3 Minuend Subtrahend Difference

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 9a 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.

Solution Slide Copyright © 2011 Pearson Education, Inc. Example 9b Write equivalent fractions with the common denominator, 8.

Slide Copyright © 2011 Pearson Education, Inc. Example 9c c – 9.03 Solution Write the equivalent addition statement – 9.03 = (– 9.03) = –4.96

Slide Copyright © 2011 Pearson Education, Inc. Example 10 In an experiment, a mixture begins at a temperature of 52.6  C. The mixture is then cooled to a temperature of  29.4  C. Find the difference between the initial and final temperatures. Solution 52.6 – (–29.4) = = 82 Answer The difference between the initial and final temperatures is 82  C.

Slide Copyright © 2011 Pearson Education, Inc. Add –6 + (–9). a) –15 b)  3 c) 3 d)

Slide Copyright © 2011 Pearson Education, Inc. Add –6 + (–9). a) –15 b)  3 c) 3 d)

Slide Copyright © 2011 Pearson Education, Inc. Subtract 5 – (–8). a) –13 b)  3 c) 3 d)

Slide Copyright © 2011 Pearson Education, Inc. Subtract 5 – (–8). a) –13 b)  3 c) 3 d)

Slide Copyright © 2011 Pearson Education, Inc. Subtract a) b) c) d) 1.3

Slide Copyright © 2011 Pearson Education, Inc. Subtract a) b) c) d) 1.3

Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Objective 1 Multiply integers.

Slide Copyright © 2011 Pearson Education, Inc. In a multiplication statement, factors are multiplied to equal a product. Product Factors

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 1 Give the name of the property of multiplication that is illustrated by each equation. a. 6(  3) =  3  6 Answer Commutative property of multiplication b. 3(  9  5) = [3(  9)]  5 Answer Associative property of multiplication c. 4(4 – 2) = 4  4 – 4  2 Answer Distributive property of multiplication over addition

Slide Copyright © 2011 Pearson Education, Inc. Multiplying Two Numbers with Different Signs When multiplying two numbers that have different signs, the product is negative.

Slide Copyright © 2011 Pearson Education, Inc. Example 2 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 © 2011 Pearson Education, Inc. Multiplying Two Numbers with the Same Sign When multiplying two numbers that have the same sign, the product is positive.

Slide Copyright © 2011 Pearson Education, Inc. Example 3 Multiply. a. –5(–9)b. (–6)(–8) Solution a. –5(–9) = b. (–6)(–8) = 45 48

Slide Copyright © 2011 Pearson Education, Inc. Objective 2 Multiply more than two numbers.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 4 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 © 2011 Pearson Education, Inc. Objective 3 Multiply rational numbers.

Slide Copyright © 2011 Pearson Education, Inc. Multiplying Fractions

Slide Copyright © 2011 Pearson Education, Inc. Example 5a Multiply Solution Divide out the common factor, 3. Because we are multiplying two numbers that have different signs, the product is negative.

Slide Copyright © 2011 Pearson Education, Inc. Example 5b Multiply Solution Divide out the common factors. Because there are an even number of negative factors, the product is positive.

Slide Copyright © 2011 Pearson Education, Inc. Multiplying Decimal Numbers To multiply decimal numbers: 1. Multiply as if they were 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 © 2011 Pearson Education, Inc. Example 6a 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 © 2011 Pearson Education, Inc. Example 6b Multiply (  3)(5.2)(1.4)(  6.1). Solution First, calculate the value and disregard signs for now. Multiply from left to right. (  3)(5.2)(1.4)(  6.1) = (15.6)(1.4)(  6.1) = 21.84(6.1) = Answer = (3)(5.2) = 15.6(1.4) The product of an even number of negative factors is positive. The factors have a total of 3 decimal places, so the product has three decimal places.

Slide Copyright © 2011 Pearson Education, Inc. Objective 4 Find the multiplicative inverse of a number.

Slide Copyright © 2011 Pearson Education, Inc. Multiplicative inverses: 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 © 2011 Pearson Education, Inc. Example 7 Find the multiplicative inverse. a.b.c.  9 Answer a. The multiplicative inverse is b. The multiplicative inverse is  8. c. The multiplicative inverse is

Slide Copyright © 2011 Pearson Education, Inc. Objective 5 Divide rational numbers.

Slide Copyright © 2011 Pearson Education, Inc. Dividend Divisor Quotient Parts of a division statement:

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 8 Divide. a. b. Solution a. b.

Slide Copyright © 2011 Pearson Education, Inc. Division Involving 0

Slide Copyright © 2011 Pearson Education, Inc. Dividing Fractions

Slide Copyright © 2011 Pearson Education, Inc. Example 9 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 © 2011 Pearson Education, Inc. Dividing Decimal Numbers To divide decimal numbers, set up a long division and consider the divisor. Case 1: If the divisor is an integer, 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, 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 © 2011 Pearson Education, Inc. Dividing Decimal Numbers continued 3. Divide the divisor into the dividend as if both numbers were whole numbers. Make sure you align the digits in the quotient properly. 4. Write the decimal point in the quotient directly above its new position in the dividend. In either case, continue the division process until you get a remainder of 0 or a repeating digit (or block of digits) in the quotient.

Slide Copyright © 2011 Pearson Education, Inc. Example 10 Divide  ÷ (  3.6) Solution Because the divisor is a decimal number, we move the decimal point enough places to the right to create an integer—in this case, one place. Then we move the decimal point one place to the right in the dividend. Because we are dividing two numbers with the same sign, the result is positive.

Slide Copyright © 2011 Pearson Education, Inc. Example 10 continued Divide  ÷ (  3.6) Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 11 Martha was decorating cookies. She used 2/3 of a container of frosting that was 3/4 full. What fractional part of the container did she use? Solution To find 2/3 of 3/4, multiply.

Slide Copyright © 2011 Pearson Education, Inc. Multiply (–6)(–3)(7). a) 126 b)  126 c) –63 d)

Slide Copyright © 2011 Pearson Education, Inc. Multiply (–6)(–3)(7). a) 126 b)  126 c) – 63 d)

Slide Copyright © 2011 Pearson Education, Inc. Divide a) b) c) d) 1.4

Slide Copyright © 2011 Pearson Education, Inc. Divide a) b) c) d) 1.4

Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Objective 1 Evaluate numbers in exponential form.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 1a 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 © 2011 Pearson Education, Inc. Example 1b Evaluate. Solution The exponent 3 means we must multiply the base by itself three times.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 2 Evaluate. a. b.c.d. Solution a. b. c. d.

Slide Copyright © 2011 Pearson Education, Inc. Objective 2 Evaluate square roots.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 3 Find all square roots of the given number. Solution a. 49 Answer  7 b.  81 Answer No real-number square roots exist.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Square Roots Involving the Radical Sign The radical symbol denotes only the positive (principal) square root.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 Evaluate the square root. a. b. c. d. Solution a. c.

Slide Copyright © 2011 Pearson Education, Inc. Objective 3 Use the order-of-operations agreement to simplify numerical expressions.

Slide Copyright © 2011 Pearson Education, Inc. Order-of- Operations Agreement Perform operations in the following order: 1. Within grouping symbols: parentheses ( ), brackets [ ], braces { }, above/below fraction bars, 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 © 2011 Pearson Education, Inc. Example 5a Simplify. Solution Divide 15 ÷ (  5) = –3 Multiply (–3)  2 = –6 Add –26 + (–6) = –32

Slide Copyright © 2011 Pearson Education, Inc. Example 5b Simplify. Solution Subtract inside the absolute value. Simplify the absolute value. Evaluate the exponent. Multiply. Add.

Slide Copyright © 2011 Pearson Education, Inc. Example 5c Simplify. Solution Calculate within the innermost parenthesis. Evaluate the exponential form, brackets, and square root. Multiply 5(3). Add Subtract 24 – 7.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 6a Simplify. Solution Subtract within the radical. Evaluate the exponential form and root. Divide. Multiply. Subtract.

Slide Copyright © 2011 Pearson Education, Inc. Sometimes fraction lines are used as grouping symbols. When they are, we simplify the numerator and denominator separately, then divide the results.

Slide Copyright © 2011 Pearson Education, Inc. Example 7a Simplify. Solution Evaluate the exponential form in the numerator and multiply in the denominator. Multiply in the numerator and subtract in the denominator. Subtract in the numerator. Divide.

Slide Copyright © 2011 Pearson Education, Inc. Example 7b Simplify. Solution Because the denominator or divisor is 0, the answer is undefined.

Slide Copyright © 2011 Pearson Education, Inc. Objective 4 Find the mean of a set of data.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 8 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 © 2011 Pearson Education, Inc. Simplify using order of operations. a)  18 b) 6 c) 30 d)

Slide Copyright © 2011 Pearson Education, Inc. Simplify using order of operations. a)  18 b) 6 c) 30 d)

Slide Copyright © 2011 Pearson Education, Inc. Simplify using order of operations. a) b) c) d) undefined 1.5

Slide Copyright © 2011 Pearson Education, Inc. Simplify using order of operations. a) b) c) d) undefined 1.5

Copyright © 2011 Pearson Education, Inc. Translating Word Phrases to Expressions Translate word phrases to expressions.

Slide Copyright © 2011 Pearson Education, Inc. Objective 1 Translating word phrases to Expressions

Slide Copyright © 2011 Pearson Education, Inc. Translating Basic Phrases AdditionTranslationSubtractionTranslation The sum of x and three x + 3The difference of x and three x – 3 h plus kh + kh minus kh – k seven added to t7 + tseven subtracted from t t – 7 three more than a number n + 3three less than a number n – 3 y increased by two y + 2y decreased by twoy – 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 © 2011 Pearson Education, Inc. Translating Basic Phrases MultiplicationTranslationDivisionTranslation The product of x and three 3x3xThe quotient of x and three x  3 or h times khkh divided by k h  k or Twice a number2n2nh divided into k k  h or Triple the number 3n3nThe ratio of a to b a  b or 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 1 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 © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 2 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 © 2011 Pearson Education, Inc. 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 

Slide Copyright © 2011 Pearson Education, Inc. 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 

Slide Copyright © 2011 Pearson Education, Inc. 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) 1.6

Slide Copyright © 2011 Pearson Education, Inc. 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) 1.6

Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Objective 1 Evaluate an expression.

Slide Copyright © 2011 Pearson Education, Inc. Evaluating an Algebraic Expression To evaluate an algebraic expression: 1. Replace the variables with their corresponding given values. 2. Calculate the numerical expression using the order of operations.

Slide Copyright © 2011 Pearson Education, Inc. Example 1a Evaluate 3w – 4(a – 6) when w = 5 and a = 7. Solution 3w – 4(a  6) 3(5) – 4(7 – 6) = 3(5) – 4(1) = 15 – 4 = 11 Replace w with 5 and a with 7. Simplify inside the parentheses first. Multiply. Subtract.

Slide Copyright © 2011 Pearson Education, Inc. Example 1b 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 © 2011 Pearson Education, Inc. Objective 2 Determine all values that cause an expression to be undefined.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 2 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 fraction will be undefined since the denominator will = 0.

Slide Copyright © 2011 Pearson Education, Inc. Objective 3 Rewrite an expression using the distributive property.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Example 3 Use the distributive property to write an equivalent expression and simplify. 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 © 2011 Pearson Education, Inc. Objective 4 Rewrite an expression by combining like terms.

Slide Copyright © 2011 Pearson Education, Inc. 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 © 2011 Pearson Education, Inc. Combining Like Terms To combine like terms, add or subtract the coefficients and keep the variables and their exponents the same.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 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 © 2011 Pearson Education, Inc. Example 5 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 © 2011 Pearson Education, Inc. Example 6 Combine like terms in 18y + 7x – y – 7x. Solution 18y + 7x – y – 7x = 17y + 0 = 17y

Slide Copyright © 2011 Pearson Education, Inc. Example 7 Combine like terms in Solution Collect like terms. Write the fraction coefficients as equivalent fractions with their LCD, 12. Combine like terms.

Slide Copyright © 2011 Pearson Education, Inc. Evaluate the expression 4(a + b) when a = 3 and b = –2. a) 4 b)  4 c) 12 d)

Slide Copyright © 2011 Pearson Education, Inc. Evaluate the expression 4(a + b) when a = 3 and b = –2. a) 4 b)  4 c) 12 d)

Slide Copyright © 2011 Pearson Education, Inc. For which values is the expression undefined? a) 8 b)  2 c)  2 and 5 d) 2 and  5 1.7

Slide Copyright © 2011 Pearson Education, Inc. For which values is the expression undefined? a) 8 b)  2 c)  2 and 5 d) 2 and  5 1.7

Slide Copyright © 2011 Pearson Education, Inc. Simplify: 7x + 8 – 2x – 4 a) 9x – 4 b) 9x + 4 c) 5x – 4 d) 5x

Slide Copyright © 2011 Pearson Education, Inc. Simplify: 7x + 8 – 2x – 4 a) 9x – 4 b) 9x + 4 c) 5x – 4 d) 5x