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4.1 Factors and Prime Factorization Goal: Write the prime factorization of a number. Vocabulary Prime number: Composite number: Prime factorization: Factor tree: Monomial: Example 1 Writing Factors A rectangle has an area of 18 square feet. Find all possible whole number dimensions of the rectangle. 1. Write 18 as a product of two whole numbers in all possible ways. p 18 The factors of 18 are The area of a rectangle can be found using the formula, Area length width. width length 62 p 18 p 18 . 2. Use the factors to find all rectangles with an area of 18 square feet that have whole number dimensions. Then label the given rectangles. Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 4.1 Factors and Prime Factorization Goal: Write the prime factorization of a number. Vocabulary A whole number greater than 1 that has exactly two whole number factors, 1 and itself Prime number: Composite number: A whole number greater than 1 that has more than two whole number factors Prime factorization: When you write a number as a product of prime numbers, you are writing its prime factorization. Factor tree: A factor tree is a diagram used to write the prime factorization of a number. Monomial: A monomial is a number, a variable, or the product of a number and one or more variables raised to whole number powers. Example 1 Writing Factors A rectangle has an area of 18 square feet. Find all possible whole number dimensions of the rectangle. 1. Write 18 as a product of two whole numbers in all possible ways. 18 p 1 18 9 p 2 18 6 p 3 18 The factors of 18 are 1, 2, 3, 6, 9, and 18 . The area of a rectangle can be found using the formula, Area length width. width length 2. Use the factors to find all rectangles with an area of 18 square feet that have whole number dimensions. Then label the given rectangles. 2 3 9 6 1 18 62 Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Writing a Prime Factorization Example 2 Write the prime factorization of 420. One possible factor tree: 420 Write original number. 10 Write 420 as 10 p p Write 10 as 6 . . Write as p6. p Write 6 as . Another possible factor tree: 420 Write original number. 6 Write 420 as 6 p p Write 6 as 10 . . Write p 10 . p Write 10 as . Both trees give the same result: 420 . Answer: The prime factorization of 420 is Example 3 as . Factoring a Monomial Factor the monomial 24x 4 y. 24x 4 y Copyright © Holt McDougal. All rights reserved. p x4y Write 24 as . p y Write x 4 as . Chapter 4 • Pre-Algebra Notetaking Guide 63 Example 2 Writing a Prime Factorization Write the prime factorization of 420. One possible factor tree: 420 10 42 Write original number. Write 420 as 10 p 42 . Write 10 as 2 p 5 . Write 42 as 2 5 7 6 2 5 7 2 3 7 p6. Write 6 as 2 p 3 . Another possible factor tree: 420 Write original number. 6 70 Write 420 as 6 p 70 . Write 6 as 2 p 3 . Write 70 as 2 3 7 10 2 3 7 2 5 7 p 10 . Write 10 as 2 p 5 . Both trees give the same result: 420 2 p 2 p 3 p 5 p 7 . Answer: The prime factorization of 420 is 22 p 3 p 5 p 7 . Example 3 Factoring a Monomial Factor the monomial 24x 4 y. 24x 4 y 2 p 2 p 2 p 3 p x 4 y Write 24 as 2 p 2 p 2 p 3 . 2 p 2 p 2 p 3 p x p x p x p x p y Write x 4 as x p x p x p x . Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 63 Checkpoint Write all factors of the number. 1. 28 2. 48 Tell whether the number is prime or composite. If it is composite, write its prime factorization. 3. 97 4. 117 Factor the monomial. 5. 21n5 64 Chapter 4 • Pre-Algebra Notetaking Guide 6. 18x 2 y 3 Copyright © Holt McDougal. All rights reserved. Checkpoint Write all factors of the number. 1. 28 2. 48 1, 2, 4, 7, 14, 28 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Tell whether the number is prime or composite. If it is composite, write its prime factorization. 3. 97 4. 117 prime composite; 32 p 13 Factor the monomial. 5. 21n5 3p7pnpnpnpnpn 64 Chapter 4 • Pre-Algebra Notetaking Guide 6. 18x 2 y 3 2p3p3pxpxpypypy Copyright © Holt McDougal. All rights reserved. 4.2 Greatest Common Factor Goal: Find the greatest common factor of two or more numbers. Vocabulary Common factor: Greatest common factor (GCF): Relatively prime: Example 1 Finding the Greatest Common Factor Volunteers A high school asks for volunteers to help clean up local highways on one Saturday each month. The group of volunteers has 27 freshman, 18 sophomores, 36 juniors, and 45 seniors. What is the greatest number of groups that can be formed if each group is to have the same number of each type of student? How many freshman, sophomores, juniors, and seniors will be in each group? Solution Method 1 List the factors of each number. Identify the greatest number that is on every list. Factors of 27: Factors of 18: Factors of 36: Factors of 45: Copyright © Holt McDougal. All rights reserved. The common factors are . The GCF is . Chapter 4 • Pre-Algebra Notetaking Guide 65 4.2 Greatest Common Factor Goal: Find the greatest common factor of two or more numbers. Vocabulary Common factor: A common factor is a whole number that is a factor of two or more nonzero whole numbers. Greatest common factor (GCF): The GCF is the greatest whole number that is a factor of two or more nonzero whole numbers. Relatively Two or more numbers are relatively prime if their greatest common factor is 1. prime: Example 1 Finding the Greatest Common Factor Volunteers A high school asks for volunteers to help clean up local highways on one Saturday each month. The group of volunteers has 27 freshman, 18 sophomores, 36 juniors, and 45 seniors. What is the greatest number of groups that can be formed if each group is to have the same number of each type of student? How many freshman, sophomores, juniors, and seniors will be in each group? Solution Method 1 List the factors of each number. Identify the greatest number that is on every list. Copyright © Holt McDougal. All rights reserved. Factors of 27: 1, 3, 9, 27 Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 45: 1, 3, 5, 9, 15, 45 The common factors are 1, 3, and 9 . The GCF is 9 . Chapter 4 • Pre-Algebra Notetaking Guide 65 Method 2 Write the prime factorization of each number. The GCF is the product of the prime factors. 27 18 The common prime factors are . 36 The GCF is . 45 Answer: The greatest number of groups that can be formed is Each group will have 27 sophomores, 36 freshman, 18 juniors, and 45 . seniors. Checkpoint Find the greatest common factor of the numbers. 1. 54, 81 Example 2 2. 12, 48, 66 Identifying Relatively Prime Numbers Find the greatest common factor of the numbers. Then tell whether the numbers are relatively prime. a. 28, 63 b. 42, 55 Solution a. List the factors of each number. Identify the greatest number that the lists have in common. Factors of 28: Factors of 63: The GCF is . So, the numbers relatively prime. b. Write the prime factorization of each number. 42 The GCF is 66 Chapter 4 • Pre-Algebra Notetaking Guide 55 . So, the numbers relatively prime. Copyright © Holt McDougal. All rights reserved. Method 2 Write the prime factorization of each number. The GCF is the product of the prime factors. 27 3 p 3 p 3 18 2 p 3 p 3 The common prime factors are 3 and 3 . 36 2 p 2 p 3 p 3 The GCF is 3 p 3 9 . 45 3 p 3 p 5 Answer: The greatest number of groups that can be formed is 9 . Each group will have 27 9 3 freshman, 18 9 2 sophomores, 36 9 4 juniors, and 45 9 5 seniors. Checkpoint Find the greatest common factor of the numbers. 1. 54, 81 2. 12, 48, 66 27 Example 2 6 Identifying Relatively Prime Numbers Find the greatest common factor of the numbers. Then tell whether the numbers are relatively prime. a. 28, 63 b. 42, 55 Solution a. List the factors of each number. Identify the greatest number that the lists have in common. Factors of 28: 1, 2, 4, 7, 14, 28 Factors of 63: 1, 3, 7, 9, 21, 63 The GCF is 7 . So, the numbers are not relatively prime. b. Write the prime factorization of each number. 42 2 p 3 p 7 The GCF is 1 . So, the numbers 66 Chapter 4 • Pre-Algebra Notetaking Guide 55 5 p 11 are relatively prime. Copyright © Holt McDougal. All rights reserved. Checkpoint Find the greatest common factor of the numbers. Then tell whether the numbers are relatively prime. 3. 30, 49 Example 3 4. 52, 78 Finding the GCF of Monomials Find the greatest common factor of 16x 2 y and 26x 2 y 3. Solution Factor the monomials. The GCF is the product of the common factors. 16x 2 y 26x 2 y 3 Answer: The GCF is . Checkpoint Find the greatest common factor of the monomials. 5. 12x 3, 18x 2 Copyright © Holt McDougal. All rights reserved. 6. 40xy 3, 24xy Chapter 4 • Pre-Algebra Notetaking Guide 67 Checkpoint Find the greatest common factor of the numbers. Then tell whether the numbers are relatively prime. 3. 30, 49 4. 52, 78 1; relatively prime Example 3 26; not relatively prime Finding the GCF of Monomials Find the greatest common factor of 16x 2 y and 26x 2 y 3. Solution Factor the monomials. The GCF is the product of the common factors. 16x 2 y 2 p 2 p 2 p 2 p x p x p y 26x 2 y 3 2 p 13 p x p x p y p y p y Answer: The GCF is 2x 2 y . Checkpoint Find the greatest common factor of the monomials. 5. 12x 3, 18x 2 6. 40xy 3, 24xy 6x 2 Copyright © Holt McDougal. All rights reserved. 8xy Chapter 4 • Pre-Algebra Notetaking Guide 67 4.3 Equivalent Fractions Goal: Write equivalent fractions. Vocabulary Equivalent fractions: Simplest form: Equivalent Fractions Words To write equivalent fractions, multiply or divide the numerator and the denominator by the same nonzero number. Algebra For all numbers a, b, and c, where b 0 and c 0, a apc a ac and . b bpc b bc 1 3 1p2 3p2 2 22 1 6 62 3 2 6 Numbers Example 1 Writing Equivalent Fractions 6 18 Write two fractions that are equivalent to . Multiply or divide the numerator and the denominator by the . 6 6p2 18 18 p 2 Multiply numerator and denominator by 2. 6 63 18 18 3 Divide numerator and denominator by 3. Answer: The fractions 68 Chapter 4 • Pre-Algebra Notetaking Guide and 6 18 are equivalent to . Copyright © Holt McDougal. All rights reserved. 4.3 Equivalent Fractions Goal: Write equivalent fractions. Vocabulary Equivalent fractions: Two fractions that represent the same number are called equivalent fractions. Simplest A fraction is in simplest form when its numerator and its denominator are relatively prime. form: Equivalent Fractions Words To write equivalent fractions, multiply or divide the numerator and the denominator by the same nonzero number. Algebra For all numbers a, b, and c, where b 0 and c 0, a apc a ac and . b bpc b bc 1 3 1p2 3p2 2 22 1 6 62 3 2 6 Numbers Example 1 Writing Equivalent Fractions 6 18 Write two fractions that are equivalent to . Multiply or divide the numerator and the denominator by the same nonzero number . 6 6p2 12 36 18 18 p 2 Multiply numerator and denominator by 2. 6 63 2 6 18 18 3 Divide numerator and denominator by 3. 12 2 6 Answer: The fractions 3 and 6 are equivalent to . 6 18 68 Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Checkpoint Write two fractions that are equivalent to the given fraction. 7 14 10 25 4 16 1. 2. Example 2 3. Writing a Fraction in Simplest Form 8 36 Write in simplest form. Write the prime factorizations of the numerator and denominator. 8 36 The GCF of 8 and 36 is 8 8 36 36 . Divide numerator and denominator by GCF. Simplify. Checkpoint Write the fraction in simplest form. 3 18 4. Copyright © Holt McDougal. All rights reserved. 12 32 5. 24 42 6. Chapter 4 • Pre-Algebra Notetaking Guide 69 Checkpoint Write two fractions that are equivalent to the given fraction. 7 14 10 25 4 16 1. 2. 1 14 , 2 28 Example 2 3. 1 8 , 4 32 2 20 , 5 50 Writing a Fraction in Simplest Form 8 36 Write in simplest form. Write the prime factorizations of the numerator and denominator. 8 23 36 2 2 p 3 2 The GCF of 8 and 36 is 22 4 . 4 8 8 36 36 4 2 9 Divide numerator and denominator by GCF. Simplify. Checkpoint Write the fraction in simplest form. 12 32 3 18 4. 1 6 Copyright © Holt McDougal. All rights reserved. 24 42 5. 6. 3 8 4 7 Chapter 4 • Pre-Algebra Notetaking Guide 69 Example 3 Simplifying a Variable Expression 14 x2 y Write in simplest form. 35x 3 14 x 2 y 35 x 3 Factor numerator and denominator. Divide out common factors. Simplify. Checkpoint Write the variable expression in simplest form. 9a 15a 7. 2 70 Chapter 4 • Pre-Algebra Notetaking Guide 16mn2 28n 8. 39s t 2 3s t 9. 2 Copyright © Holt McDougal. All rights reserved. Example 3 Simplifying a Variable Expression 14 x2 y Write in simplest form. 35x 3 2p7pxpxpy 14 x 2 y 3 35 x 5p7pxpxpx 1 1 1 2p7 p xp xpy 5p7 p xp xpx 1 2y 5 x 1 Factor numerator and denominator. Divide out common factors. 1 Simplify. Checkpoint Write the variable expression in simplest form. 16mn2 28n 9a 15a 7. 2 8. 3 5a 70 Chapter 4 • Pre-Algebra Notetaking Guide 4mn 7 39s t 2 3s t 9. 2 13t s Copyright © Holt McDougal. All rights reserved. 4.4 Least Common Multiple Goal: Find the least common multiple of two numbers. Vocabulary Multiple: Common multiple: Least common multiple (LCM): Least common denominator (LCD): Example 1 Finding the Least Common Multiple Find the least common multiple of 6 and 14. Solution You can use one of two methods to find the LCM. Method 1 List the multiples of each number. Identify the least number that is on both lists. The LCM of 6 and 14 is . Multiples of 6: Multiples of 14: Method 2 Find the common factors of the numbers. 6 14 The common factor is . Multiply all of the factors, using each common factor only once. LCM Answer: Both methods get the same result. The LCM is Copyright © Holt McDougal. All rights reserved. . Chapter 4 • Pre-Algebra Notetaking Guide 71 4.4 Least Common Multiple Goal: Find the least common multiple of two numbers. Vocabulary Multiple: A multiple of a whole number is the product of the number and any nonzero whole number. Common A multiple that is shared by two or more numbers multiple: is a common multiple. Least common The least common multiple of two or more multiple (LCM): numbers The least common multiple of the Least common denominator (LCD): denominators of two or more fractions Example 1 Finding the Least Common Multiple Find the least common multiple of 6 and 14. Solution You can use one of two methods to find the LCM. Method 1 List the multiples of each number. Identify the least number that is on both lists. Multiples of 6: Multiples of 14: 6, 12, 18, 24, 30, 36, 42, 48 14, 28, 42, 56 The LCM of 6 and 14 is 42 . Method 2 Find the common factors of the numbers. 6 2p3 14 2 p 7 The common factor is 2 . Multiply all of the factors, using each common factor only once. LCM 2 p 3 p 7 42 Answer: Both methods get the same result. The LCM is 42 . Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 71 Example 2 Finding the Least Common Multiple of Monomials Find the least common multiple 6xy and 16x 2. 6xy 16x 2 LCM Answer: The least common multiple of 6xy and 16x 2 is . Checkpoint Find the least common multiple of the numbers or the monomials. 1. 8, 18 2. 4, 5, 15 3. 12x, 18x 2 4. 4xy, 10xz 2 Example 3 Comparing Fractions Using the LCD Summer Sports Last year, a summer resort had 165,000 visitors, including 44,000 water skiers. This year, the resort had 180,000 visitors, including 63,000 water skiers. In which year was the fraction of water skiers greater? Solution 1. Write the fractions and simplify. Number of water skiers Total number of visitors Number of water skiers Total number of visitors Last year: This year: 2. Find the LCD of and . The LCM of . So, the LCD of the fractions is 72 Chapter 4 • Pre-Algebra Notetaking Guide and is . Copyright © Holt McDougal. All rights reserved. Example 2 Finding the Least Common Multiple of Monomials Find the least common multiple 6xy and 16x 2. 6xy 2 p 3 p x p y 16x 2 2 p 2 p 2 p 2 p x p x LCM 2 p 2 p 2 p 2 p 3 p x p x p y 48x 2y Answer: The least common multiple of 6xy and 16x 2 is 48x 2y . Checkpoint Find the least common multiple of the numbers or the monomials. 1. 8, 18 2. 4, 5, 15 72 60 3. 12x, 18x 2 4. 4xy, 10xz 2 36x 2 Example 3 20xyz 2 Comparing Fractions Using the LCD Summer Sports Last year, a summer resort had 165,000 visitors, including 44,000 water skiers. This year, the resort had 180,000 visitors, including 63,000 water skiers. In which year was the fraction of water skiers greater? Solution 1. Write the fractions and simplify. Number of water skiers 44,000 4 Number of water skiers 63,000 7 Last year: 165,000 15 Total number of visitors This year: 180,000 20 Total number of visitors 2. Find the LCD of 4 and 7 . The LCM of 15 and 20 is 15 20 60 . So, the LCD of the fractions is 60 . 72 Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 3. Write equivalent fractions using the LCD. Last year: This year: < 4. Compare the numerators: , so < . Answer: The fraction of water skiers was greater . Ordering Fractions and Mixed Numbers Example 4 5 9 12 2 33 8 Order the numbers 4 , , and from least to greatest. 1. Write the mixed number as an improper fraction. 5 12 4 12 12 9 2 33 8 2. Find the LCD of , , and . The LCM of 12, 2, 12 and 8 is . So, the LCD is . 3. Write equivalent fractions using the LCD. p 9p 9 2p 12 p 12 2 33 p 33 8 p 8 < 4. Compare the numerators: < so < and and < , . Answer: From least to greatest, the numbers are , Copyright © Holt McDougal. All rights reserved. , and . Chapter 4 • Pre-Algebra Notetaking Guide 73 3. Write equivalent fractions using the LCD. Last year: 16 4 4p4 60 15 15 p 4 This year: 21 7 7p3 60 20 20 p 3 4. Compare the numerators: 21 16 < , so 60 60 4 7 < . 15 20 Answer: The fraction of water skiers was greater this year . Example 4 Ordering Fractions and Mixed Numbers 5 9 12 2 33 8 Order the numbers 4 , , and from least to greatest. 1. Write the mixed number as an improper fraction. 53 4 p 12 5 5 4 12 12 12 53 9 33 2. Find the LCD of , , and . The LCM of 12, 2, 8 12 2 and 8 is 24 . So, the LCD is 24 . 3. Write equivalent fractions using the LCD. 53 p 2 12 53 9p 106 108 9 24 24 2 p 12 12 p 2 12 2 3 33 p 33 8 p 3 8 99 24 4. Compare the numerators: so 33 8 5 < 4 12 5 99 24 and 4 1 < 2 106 < 24 9 2 106 108 and 2 < , 4 24 . Answer: From least to greatest, the numbers are 33 5 , 4 , and 8 12 Copyright © Holt McDougal. All rights reserved. 9 2 . Chapter 4 • Pre-Algebra Notetaking Guide 73 4.5 Rules of Exponents Goal: Multiply and divide powers. Product of Powers Property Words To multiply powers with the same base, add their exponents. Algebra a m p a n a m n Numbers 43 p 42 4 4 Using the Product of Powers Property Example 1 a. 47 p 411 4 4 b. 2x 2 p 7x 6 2 p 7 p x 2 p x 6 Product of powers property Add exponents. Commutative property of multiplication 2p7px Product of powers property 2p7px Add exponents. Multiply. Checkpoint Find the product. Write your answer using exponents. 74 1. 25 p 212 2. (0.4)6 • (0.4)2 • (0.4)3 3. x 6 p x 13 4. b2 p b4 p b Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 4.5 Rules of Exponents Goal: Multiply and divide powers. Product of Powers Property Words To multiply powers with the same base, add their exponents. Algebra a m p a n a m n Numbers 43 p 42 4 32 4 5 Using the Product of Powers Property Example 1 a. 47 p 411 4 4 7 11 Product of powers property 18 b. 2x 2 p 7x 6 2 p 7 p x 2 p x 6 2p7px 26 2p7px 8 14x 8 Add exponents. Commutative property of multiplication Product of powers property Add exponents. Multiply. Checkpoint Find the product. Write your answer using exponents. 1. 25 p 212 2. (0.4)6 • (0.4)2 • (0.4)3 217 3. x 6 p x 13 4. b2 p b4 p b x 19 74 Chapter 4 • Pre-Algebra Notetaking Guide (0.4)11 b7 Copyright © Holt McDougal. All rights reserved. Quotient of Powers Property Words To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. am a mn , where a 0 Algebra n a 57 5 Numbers 4 5 Using the Quotient of Powers Property Example 2 (0.6)8 a. (0.6 (0.6) )3 (0.6) 3x7 12x 5 Quotient of powers property Subtract exponents. 3x b. 3 12 3x 12 Quotient of powers property Subtract exponents. Divide numerator and denominator by . Checkpoint Find the quotient. Write your answer using exponents. 59 5 (1.4)7 6. 4 4x 13 24 x 8. 11 5. 2 7. 9 Copyright © Holt McDougal. All rights reserved. (1.4) 14 x 16 6x Chapter 4 • Pre-Algebra Notetaking Guide 75 Quotient of Powers Property Words To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. am a mn , where a 0 Algebra n a 57 5 Numbers 4 5 Example 2 74 5 3 Using the Quotient of Powers Property (0.6)8 83 a. (0.6 Quotient of powers property 3 (0.6) ) 5 (0.6) Subtract exponents. 73 3x7 3x b. 3 12x 12 Quotient of powers property 4 3x 12 Subtract exponents. 4 x 4 Divide numerator and denominator by 3 . Checkpoint Find the quotient. Write your answer using exponents. 59 5 (1.4)7 6. 4 5. 2 (1.4) 57 14 x 16 6x 4x 13 24 x 7. 9 8. 11 x4 6 Copyright © Holt McDougal. All rights reserved. (1.4)3 7x 5 3 Chapter 4 • Pre-Algebra Notetaking Guide 75 Using Both Properties of Powers Example 3 4m3 p m4 12m Simplify 2 . 4m3 p m4 4m 2 12m 12m2 4m 12m 2 4m 12 4m 12 Product of powers property Add exponents. Quotient of powers property Subtract exponents. Divide numerator and denominator by . Checkpoint Simplify. 6m5 p m 15m 9. 3 76 Chapter 4 • Pre-Algebra Notetaking Guide n2 p 10n6 5n 10. 3 Copyright © Holt McDougal. All rights reserved. Example 3 Using Both Properties of Powers 4m3 p m4 12m Simplify 2 . 34 4m3 p m4 4m 2 12m 12m2 Product of powers property 7 4m 12m 2 Add exponents. 72 4m 12 Quotient of powers property 5 4m 12 5 m 3 Subtract exponents. Divide numerator and denominator by 4 . Checkpoint Simplify. n2 p 10n6 5n 6m5 p m 15m 9. 3 10. 3 2m 3 5 76 Chapter 4 • Pre-Algebra Notetaking Guide 2n 5 Copyright © Holt McDougal. All rights reserved. 4.6 Negative and Zero Exponents Goal: Work with negative and zero exponents. Negative and Zero Exponents For any nonzero number a, a0 1. 1 a For any nonzero number a and any integer n, an n . Powers with Negative and Zero Exponents Example 1 Write the expression using only positive exponents. a. 43 Definition of negative exponent b. m5n0 m5 p Definition of zero exponent Definition of negative exponent c. 13xy8 Definition of negative exponent Checkpoint Write the expression using only positive exponents. 1. 33,3330 Example 2 2. 73 3. 2z2 4. 3x 4 y Rewriting Fractions Write the expression without using a fraction bar. a. 1 15 Definition of negative exponent a3 c Definition of negative exponent b. 5 Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 77 4.6 Negative and Zero Exponents Goal: Work with negative and zero exponents. Negative and Zero Exponents For any nonzero number a, a0 1. 1 a For any nonzero number a and any integer n, an n . Powers with Negative and Zero Exponents Example 1 Write the expression using only positive exponents. 1 a. 43 3 Definition of negative exponent b. m5n0 m5 p 1 Definition of zero exponent 4 1 5 Definition of negative exponent m 13x c. 13xy8 8 Definition of negative exponent y Checkpoint Write the expression using only positive exponents. 2. 73 1. 33,3330 3. 2z2 1 1 3 7 Example 2 Rewriting Fractions 4. 3x 4 y 2 2 z 3y 4 x Write the expression without using a fraction bar. 1 15 Definition of negative exponent a3 c Definition of negative exponent a. 151 b. 5 a 3c5 Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 77 Checkpoint Write the expression without using a fraction bar. 1 18 1 100 5. Example 3 x5 y 3 c 6. 7. 2 8. 7 Using Powers Properties with Negative Exponents Find the product or quotient. Write your answer using only positive exponents. 0.7n4 b. a. 612 p 64 n Solution a. 612 p 64 6 6 0.7n4 b. 0.7n n Product of powers property Add exponents. Quotient of powers property 0.7n Subtract exponents. Definition of negative exponent Checkpoint Find the product or quotient. Write your answer using only positive exponents. 9. (0.3)10 p (0.3)7 78 Chapter 4 • Pre-Algebra Notetaking Guide 7d 4 d 10. 2 Copyright © Holt McDougal. All rights reserved. Checkpoint Write the expression without using a fraction bar. 1 18 1 100 5. 181 Example 3 x5 y 3 c 6. 7. 2 1001 or 102 8. 7 3c2 x 5 y 7 Using Powers Properties with Negative Exponents Find the product or quotient. Write your answer using only positive exponents. 0.7n4 b. a. 612 p 64 n Solution a. 612 p 64 6 6 12 (4) 8 0.7n4 b. 0.7n n 0.7n 5 0.7 n 5 Product of powers property Add exponents. 4 1 Quotient of powers property Subtract exponents. Definition of negative exponent Checkpoint Find the product or quotient. Write your answer using only positive exponents. 9. (0.3)10 p (0.3)7 (0.3)3 78 Chapter 4 • Pre-Algebra Notetaking Guide 7d 4 d 10. 2 7 6 d Copyright © Holt McDougal. All rights reserved. 4.7 Scientific Notation Goal: Write numbers using scientific notation. Using Scientific Notation A number is written in scientific notation if it has the form c 10n where 1 ≤ c < 10 and n is an integer. Standard form Product form Scientific notation 725,000 7.25 100,000 7.25 10 5 0.006 6 0.001 6 103 Example 1 Writing Numbers in Scientific Notation a. The average distance Mars is from the sun is 141,600,000 miles. Write this number in scientific notation. Standard form Product form Scientific notation b. The diameter of a quarter-ounce gold American Eagle coin is 0.022 meter. Write this number in scientific notation. Standard form Product form Scientific notation Example 2 Writing Numbers in Standard Form a. Write 4.1 104 in standard form. Scientific notation Product form Standard form b. Write 7.23 106 in standard form. Scientific notation Product form Standard form Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 79 4.7 Scientific Notation Goal: Write numbers using scientific notation. Using Scientific Notation A number is written in scientific notation if it has the form c 10n where 1 ≤ c < 10 and n is an integer. Standard form Product form Scientific notation 725,000 7.25 100,000 7.25 10 5 0.006 6 0.001 6 103 Example 1 Writing Numbers in Scientific Notation a. The average distance Mars is from the sun is 141,600,000 miles. Write this number in scientific notation. Standard form Product form Scientific notation 141,600,000 1.416 100,000,000 1.416 108 b. The diameter of a quarter-ounce gold American Eagle coin is 0.022 meter. Write this number in scientific notation. Standard form Product form Scientific notation Example 2 2.2 102 2.2 0.01 0.022 Writing Numbers in Standard Form a. Write 4.1 104 in standard form. Scientific notation Product form 4.1 104 4.1 10,000 b. Write 7.23 106 in standard form. Scientific notation Product form 7.23 10 6 Copyright © Holt McDougal. All rights reserved. 7.23 0.000001 Standard form 41,000 Standard form 0.00000723 Chapter 4 • Pre-Algebra Notetaking Guide 79 Checkpoint Write the number in scientific notation. 1. 3,050,000,000 2. 0.000082 Write the number in standard form. 4. 9.2 104 3. 6.53 107 Example 3 Ordering Numbers Using Scientific Notation Order 5.3 105, 520,000, and 7.5 104 from least to greatest. 1. Write each number in scientific notation if necessary. 520,000 2. Order the numbers with different powers of 10. < 10 Because 10 < , < and . 3. Order the numbers with the same power of 10. Because < , < . 4. Write the original numbers in order from least to greatest. ; ; Checkpoint Order the numbers from least to greatest. 5. 23,000; 3.4 103; 2.2 104 6. 4.5 104; 0.000047; 4.8 105 80 Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Checkpoint Write the number in scientific notation. 1. 3,050,000,000 2. 0.000082 8.2 105 3.05 10 9 Write the number in standard form. 4. 9.2 104 3. 6.53 107 65,300,000 Example 3 0.00092 Ordering Numbers Using Scientific Notation Order 5.3 105, 520,000, and 7.5 104 from least to greatest. 1. Write each number in scientific notation if necessary. 520,000 5.2 10 5 2. Order the numbers with different powers of 10. Because 10 4 < 10 5 , 7.5 10 4 < 5.3 10 5 and 7.5 10 4 < 5.2 10 5 . 3. Order the numbers with the same power of 10. Because 5.2 < 5.3 , 5.2 10 5 < 5.3 10 5 . 4. Write the original numbers in order from least to greatest. 7.5 10 4 ; 520,000 ; 5.3 10 5 Checkpoint Order the numbers from least to greatest. 5. 23,000; 3.4 103; 2.2 104 3.4 103; 2.2 10 4; 23,000 6. 4.5 104; 0.000047; 4.8 105 0.000047; 4.8 105; 4.5 104 80 Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Multiplying Numbers in Scientific Notation Example 4 Oxygen Atoms The volume of one mole of oxygen atoms is about 1.736 105 cubic meters. Find the volume of 1.5 104 moles of oxygen atoms. Solution Number of Volume of one mole Total moles of oxygen atoms volume ( )( ( ) )( ( ( Substitute values. ) ) ) Commutative and associative properties of multiplication Multiply and . Product of powers property Add exponents. Answer: The volume of 1.5 104 moles of oxygen atoms is about cubic meters. Checkpoint Find the product. Write your answer in scientific notation. 7. (2.5 103)(2 105) Copyright © Holt McDougal. All rights reserved. 8. (1.5 102)(4 104) Chapter 4 • Pre-Algebra Notetaking Guide 81 Multiplying Numbers in Scientific Notation Example 4 Oxygen Atoms The volume of one mole of oxygen atoms is about 1.736 105 cubic meters. Find the volume of 1.5 104 moles of oxygen atoms. Solution Number of Volume of one mole Total moles of oxygen atoms volume ( 1.736 105 )( 1.5 104 ) ( 1.736 1.5 )( 10 5 10 2.604 ( 105 104 ) 2.604 ( 105 4 ) 2.604 10 1 4 Substitute values. ) Commutative and associative properties of multiplication Multiply 1.736 and 1.5 . Product of powers property Add exponents. Answer: The volume of 1.5 104 moles of oxygen atoms is about 2.604 10 1 cubic meters. Checkpoint Find the product. Write your answer in scientific notation. 7. (2.5 103)(2 105) 5 108 Copyright © Holt McDougal. All rights reserved. 8. (1.5 102)(4 104) 6 106 Chapter 4 • Pre-Algebra Notetaking Guide 81 4 Words to Review Give an example of the vocabulary word. 82 Prime number Composite number Prime factorization Factor tree Monomial Common factor Greatest common factor (GCF) Relatively prime Chapter 4 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 4 Words to Review Give an example of the vocabulary word. Prime number 3 Prime factorization 62p3 Composite number 8 Factor tree 420 10 42 2 5 7 6 2 5 7 2 3 Monomial 2x Greatest common factor (GCF) The GCF of 12 and 18 is 6. 82 Chapter 4 • Pre-Algebra Notetaking Guide Common factor 2 is a common factor of 4 and 6. Relatively prime 10 and 11 are relatively prime. Copyright © Holt McDougal. All rights reserved. Equivalent fractions Simplest form Multiple Common multiple Least common multiple (LCM) Least common denominator (LCD) Scientific notation Review your notes and Chapter 4 by using the Chapter Review on pages 212–215 of your textbook. Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 83 Equivalent fractions 12 36 Simplest form 2 6 The fractions and 6 18 are equivalent to . Multiple 12 is a multiple of 6. Least common multiple (LCM) The LCM of 12 and 18 is 36. 12 written in simplest form 36 1 is . 3 Common multiple 10 is a common multiple of 5 and 2. Least common denominator (LCD) 5 12 7 18 The LCD of and is 36. Scientific notation 1.2 103 Review your notes and Chapter 4 by using the Chapter Review on pages 212–215 of your textbook. Copyright © Holt McDougal. All rights reserved. Chapter 4 • Pre-Algebra Notetaking Guide 83