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Unit 1: Laws of Exponents and Scientific Notation Name Date Period Math 8: Mr. Sanford Lesson 1 8•1 Lesson 1: Powers of 10 Exercise 1 100 10 Exercise 2 5 Write 10 as a product of tens, then find the express the product in standard form (as a single number). Exercise 3 Write 10 × 100 as a product of tens, then find the product in standard form. Exercise 4 Write 10 × 100 as a product of tens, then find the product in standard form. Exercise 5 Write 3 × 102 without using an exponent, then find the product. Exercise 6 Find the product of each of the following. a. 3.4 × 103 b. c. 54.18 × 105 . 035 × 108 Lesson 1: Powers of 10 1 Lesson 1 8•1 Exercise 7 Write 700 ÷ 102 without using an exponent, then find the quotient. Exercise 8 Write 7.1 ÷ 102 without using an exponent, then find the quotient. Exercise 9 Complete the pattern below. Describe what is happening. 0.043; 4.3; 430; __________; __________; __________ Exercise 10 Complete the pattern below. Describe what is happening. 6,300,000; __________; 630; 6.3; _______; ___________ Lesson 1: Powers of 10 2 Lesson 1 8•1 Problem Set 1. Write the following in exponential form (e.g., 100 = 102). a. 10,000 = d. 100 x 100 = b. 1000 e. 1,000,000 = = c. 10 x 10 = f. 1000 × 1000 = 2. Write the following in standard form (e.g., 5 x 102= 500). a. 9 x 103 = e. 4.025 x 103 = b. 39 x 104 = f. 40.25 x 104 = c. 7200 ÷ 102 = g. 725 ÷ 103 = d. 7,200,000 ÷ 103 = h. 7.2 ÷ 102 = 3. Complete the patterns. a. 0.03 0.3 b. 6,500,000 ________ 65,000 c. _________ 9,430 d. 999 99,900 9,990 Lesson 1: 30 _____________ ___________ 6.5 _________ 94.3 __________ Powers of 10 _____________ _____________ 9.43 __________ __________ ___________ 3 Lesson 2 8•1 Lesson 2: Exponential Notation 𝟗 𝟒 𝟓𝟔 means 𝟓 × 𝟓 × 𝟓 × 𝟓 × 𝟓 × 𝟓 and � � means 𝟕 𝟗 𝟕 × 𝟗 𝟕 × 𝟗 𝟕 𝟗 × . 𝟕 You have seen this kind of notation before, it is called exponential notation. In general, for any number 𝒙 and any positive integer 𝒏, (𝒙 ��� 𝒙𝒏 = �� ∙ 𝒙 ⋯�� 𝒙) 𝒏 𝒕𝒊𝒎𝒆𝒔 𝒏 The number 𝒙 is called 𝒙 raised to the 𝒏-th power, 𝒏 is the exponent of 𝒙 in 𝒙𝒏 and 𝒙 is the base of 𝒙𝒏 . Exercise 1 Exercise 6 4 ×��� �� ⋯× �� 4= 7 7 × ⋯× = �� 2 ����� 2 Exercise 2 Exercise 7 3.6�×��� �� ⋯ ×��� 3.6 = 3.647 (−13) × ⋯ × (−13) = ������������� Exercise 3 Exercise 8 (−11.63) × ⋯ × (−11.63) = ����������������� 1 1 �− � × ⋯ × �− � = ��������������� 14 14 Exercise 4 Exercise 9 7 𝑡𝑖𝑚𝑒𝑠 21 𝑡𝑖𝑚𝑒𝑠 6 𝑡𝑖𝑚𝑒𝑠 _______ 𝑡𝑖𝑚𝑒𝑠 34 𝑡𝑖𝑚𝑒𝑠 10 𝑡𝑖𝑚𝑒𝑠 15 12 � �� ×��� ⋯× ��� 12 = 12 𝑥 ∙ 𝑥⋯𝑥 = ����� Exercise 5 Exercise 10 (−5) × ⋯ × (−5) = ����������� 𝑥 ∙ 𝑥 ⋯ 𝑥 = 𝑥𝑛 ����� _______𝑡𝑖𝑚𝑒𝑠 185 𝑡𝑖𝑚𝑒𝑠 10 𝑡𝑖𝑚𝑒𝑠 _______𝑡𝑖𝑚𝑒𝑠 Lesson 2: Exponential Notation 4 Lesson 2 8•1 Exercise 11 Will these products be positive or negative? How do you know? (−1) × (−1) × ⋯ × (−1) = (−1)12 ����������������� 12 𝑡𝑖𝑚𝑒𝑠 (−1) × (−1) × ⋯ × (−1) = (−1)13 ����������������� 13 𝑡𝑖𝑚𝑒𝑠 Exercise 12 Is it necessary to do all of the calculations to determine the sign of the product? Why or why not? (−5) × (−5) × ⋯ × (−5) = (−5)95 ����������������� 95 𝑡𝑖𝑚𝑒𝑠 (−1.8) × (−1.8) × ⋯ × (−1.8) = (−1.8)122 ��������������������� 122 𝑡𝑖𝑚𝑒𝑠 Exercise 13 Fill in the blanks about whether the number is positive or negative. If 𝑛 is a positive even number, then (−55)𝑛 is __________________________. If 𝑛 is a positive odd number, then (−72.4)𝑛 is __________________________. Lesson 2: Exponential Notation 5 Lesson 2 8•1 Problem Set 1. Use what you know about exponential notation to complete the expressions below. (−𝟓) × ⋯ × (−𝟓) = ����������� 𝟑. 𝟕�� �� ×��� ⋯× ��𝟑. �� 𝟕 = 𝟑. 𝟕𝟏𝟗 𝟕 ×��� �� ⋯× �� 𝟕 = 𝟕𝟒𝟓 𝟔 ×��� �� ⋯× �� 𝟔= 𝟒. 𝟑�� �� ×��� ⋯× ��𝟒. �� 𝟑= (−𝟏. 𝟏) × ⋯ × (−𝟏. 𝟏) = ��������������� 𝟐 𝟐 � � × ⋯× � � = �� 𝟑 ��������� 𝟑 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝒙 �− � × ⋯ × �− � = �− � ��������������� 𝟓 𝟓 𝟓 (−𝟏𝟐) × ⋯ × (−𝟏𝟐) = (−𝟏𝟐)𝟏𝟓 ������������� 𝒂 ×��� �� ⋯× �� 𝒂= _____ 𝒕𝒊𝒎𝒆𝒔 𝟏𝟕 𝒕𝒊𝒎𝒆𝒔 𝟒 𝒕𝒊𝒎𝒆𝒔 _____ 𝒕𝒊𝒎𝒆𝒔 𝟏𝟑 𝒕𝒊𝒎𝒆𝒔 𝟗 𝒕𝒊𝒎𝒆𝒔 𝟏𝟗 𝒕𝒊𝒎𝒆𝒔 _____ 𝒕𝒊𝒎𝒆𝒔 𝒎 𝒕𝒊𝒎𝒆𝒔 _____ 𝒕𝒊𝒎𝒆𝒔 2. Write an expression with (−1) as its base that will produce a positive product. 3. Write an expression with (−1) as its base that will produce a negative product. 4. Rewrite each number in exponential notation using 2 as the base. 8= 16 = 32 = 64 = 128 = 256 = 5. Could −2 be used as a base to rewrite 32? 64? Why or why not? Lesson 2: Exponential Notation 6 Lesson 3 8•1 Lesson 3: Multiplication of Numbers in Exponential Form Notes: In general, if 𝑥 is any number and 𝑚, 𝑛 are positive integers, then 𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛 because 𝑥 𝑚 × 𝑥 𝑛 = (𝑥 ⋯�𝑥) ⋯�𝑥) ⋯�𝑥) ��� � × (𝑥 ��� � = (𝑥 ��� � = 𝑥 𝑚+𝑛 𝑚 𝑡𝑖𝑚𝑒𝑠 𝑛 𝑡𝑖𝑚𝑒𝑠 𝑚+𝑛 𝑡𝑖𝑚𝑒𝑠 Exercise 1 Exercise 5 𝟏𝟒𝟐𝟑 × 𝟏𝟒𝟖 = Let 𝑎 be a number. Exercise 2 Exercise 6 𝑎23 ∙ 𝑎8 = (−72)10 × (−72)13 = Let f be a number. Exercise 3 Exercise 7 𝑓 10 ∙ 𝑓 13 = 𝟓𝟗𝟒 × 𝟓𝟕𝟖 = Let 𝑏 be a number. Exercise 4 Exercise 8 𝒃𝟗𝟒 ∙ 𝒃𝟕𝟖 = Let 𝑥 be a positive integer. If (−3)9 × (−3) 𝑥 = (−3)14 , what is 𝑥? (−𝟑)𝟗 × (−𝟑)𝟓 = Lesson 3: Multiplication of Numbers in Exponential Form 7 Lesson 3 Exercise 9 Exercise 11 65 × 49 × 43 × 614 = 24 × 82 = 24 × 26 = Exercise 10 Exercise 12 (−4)2 ∙ 175 ∙ (−4)3 ∙ 177 = 37 × 9 = 8•1 Exercise 13 Let 𝑎 and 𝑏 be numbers. Use an array to simplify the expression, 𝑎(𝑎 + 𝑏) = Exercise 14 Let 𝑎 and 𝑏 be numbers. Use an array to simplify the expression, 𝑏(𝑎 + 𝑏) = Exercise 15 Let 𝑎 and 𝑏 be numbers. Use an array to simplify the expression, (𝑎 + 𝑏)(𝑎 + 𝑏) = Lesson 3: Multiplication of Numbers in Exponential Form 8 Lesson 3 8•1 Problem Set 1. Rewrite the rule for multiplying number with a common base. 2. Multiply each of the following: a. (−19)5 ∙ (−19)11 = b. 2.75 × 2.73 = 1 𝑎 5 1 𝑏 5 c. � � ∙ � � = d. 𝑐 3 ∙ 𝑐 5 = e. 𝑎10 (−𝑏)5 ∙ 𝑎4 (−𝑏)8 = f. 1 5 ℎ 1 6 ℎ � � ∙� � = g. 6𝑠 6 ∙ 8𝑠 3 ∙ 3𝑠 5 = h. 2𝑘𝑧 5 ∙ 4𝑘 2 𝑧 6 = Lesson 3: Multiplication of Numbers in Exponential Form 9 Lesson 4 8•1 Lesson 4: Division of Numbers in Exponential Form Notes: In general, if 𝑥 is nonzero and 𝑚, 𝑛 are positive integers, then 𝑥𝑚 = 𝑥 𝑚−𝑛 𝑥𝑛 𝑖𝑓 𝑚 > 𝑛 Directions: Simplify any expressions below that can be simplified. If they cannot be simplified, explain why not. Exercise 1 9 8 9 � � 5 = 8 2 � � 5 7 = 76 Exercise 2 (−5)16 (−5)7 Exercise 3 = Exercise 4 135 = 134 Exercise 5 Lesson 8 27 = 24 35 ∙ 28 = 3 2 ∙ 23 Exercise 7 Lesson 9 323 = 33 (−2)7 ∙ 955 = (−2)5 ∙ 954 Lesson 4: Division of numbers in Exponential Form 10 Lesson 4 8•1 Problem Set 1. Rewrite the rule for dividing numbers with a common base. 2. Divide each of the following: a. 710 b. 1222 c. � � d. 𝑦 3𝑧 5 e. (−3)4 f. 73 = 1210 = 2 15 3 2 9 � � 3 𝑦 2𝑧 4 (−3)3 48 813 810 46 = = = = Lesson 4: Division of numbers in Exponential Form 11 Lesson 5 8•1 Lesson 5: Numbers in Exponential Form Raised to a Power Notes: For any number 𝑥 and any positive integers 𝑚 and 𝑛, because (𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛 (𝑥 𝑚 )𝑛 = ������� (𝑥 ∙ 𝑥 ⋯ 𝑥)𝑛 𝑚 𝑡𝑖𝑚𝑒𝑠 (𝑥 ��� (𝑥 ��� = �� ∙ 𝑥 ⋯�� 𝑥) × ⋯ × �� ∙ 𝑥 ⋯�� 𝑥) =𝑥 Exercise 1 𝑚 𝑡𝑖𝑚𝑒𝑠 𝑚𝑛 𝑚 𝑡𝑖𝑚𝑒𝑠 Exercise 3 (153 )9 = (3.417 )4 = Exercise 2 Exercise 4 ((−2)5 )8 Let 𝑠 be a number. (𝑠17 )4 = = (𝑛 𝑡𝑖𝑚𝑒𝑠) Exercise 5 Sarah wrote that (35 )7 = 312 . Correct her mistake. Write an exponential expression using a base of 3 and exponents of 5, 7, and 12 that would make her answer correct. Lesson 5: Numbers in Exponential Form Raised to a Power 12 Lesson 5 8•1 Notes: For any numbers 𝑥 and 𝑦, and positive integer 𝑛, (𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛 because (𝑥𝑦)𝑛 = (𝑥𝑦) ⋯ (𝑥𝑦) ������ ��� 𝑛 𝑡𝑖𝑚𝑒𝑠 (𝑦 ��� = (𝑥 ∙ 𝑥 ⋯�� 𝑥) ∙ �� ∙ 𝑦 ⋯�� 𝑦) ����� 𝑛 𝑡𝑖𝑚𝑒𝑠 𝑛 𝑛 =𝑥 𝑦 𝑛 𝑡𝑖𝑚𝑒𝑠 Exercise 6 Exercise 9 (11 × 4)9 = Let 𝑥 be a number. (5𝑥)7 = Exercise 7 Exercise 10 (32 ×7 4 )5 Let 𝑥 and 𝑦 be numbers. (5𝑥𝑦 2 )7 = = Exercise 8 Exercise 11 Let 𝑎, 𝑏, and 𝑐 be numbers. (32 𝑎4 )5 = Let 𝑎, 𝑏, and 𝑐 be numbers. (𝑎2 𝑏𝑐 3 )4 = Exercise 13 Exercise 13 𝑥 𝑛 � � = 𝑦 4 3 � � = 5 Lesson 5: Numbers in Exponential Form Raised to a Power 13 Lesson 5 8•1 Problem Set Exponent Laws 1. Multiplying same base: 𝑑 4 ∙ 𝑑 8 = 𝑑 4+8 = 𝑑12 2. Dividing same base: 𝑡6 𝑡4 = 𝑡 6−4 = 𝑡 2 Add exponents Subtract exponents (top – bottom) 3. Powers of exponents: (𝑎3 )5 = 𝑎3∙5 = 𝑎15 Multiply exponents 4. Powers of products: (𝑎2 𝑏6 )4 = 𝑎2∙4 𝑏6∙4 = 𝑎8 𝑏24 Multiply each exponent Directions: Simplify each expression using the exponent laws above and state which law you used. 1. (53 )4 = 2. 310 ∙ 35 = 3. (85 × 62 )6 = 4. 𝑥 4 ∙ 𝑥 4 = 5. (3𝑐 3 )5 = 6. 47 43 = 𝑟 10 7. � 3 � = 𝑟6 𝑥 5 𝑦5 3 8. �𝑦4 ∙ 𝑥 4 � = Lesson 5: Numbers in Exponential Form Raised to a Power 14 Lesson 6 8•1 Lesson 6: Negative and Zero Exponents Notes: Exercise 1 Write the expanded form of 8,374 using the exponential notation. Exercise 2 Write the expanded form of 6,985,062 using the exponential notation. Exercise 3 What is the value of (3 × 10−2 )? Lesson 6: Negative and Zero Exponents 15 Lesson 6 8•1 Exercise 4 What is the value of (3 × 10−5 )? Exercise 5 Write the complete expanded form of the decimal 4.728 in exponential notation. For Exercises 5–10, write an equivalent expression, in exponential notation, to the one given and simplify as much as possible. Exercise 5 Exercise 8 5−3 = Let 𝑥 be a nonzero number. Exercise 6 Exercise 9 1 = 89 Let 𝑥 be a nonzero number. Exercise 7 Exercise 10 3 ∙ 2−4 = Let 𝑥, 𝑦 be two nonzero numbers. 𝑥 −3 = 1 = 𝑥9 𝑥𝑦 −4 = Lesson 6: Negative and Zero Exponents 16 Lesson 6 8•1 Problem Set Let 𝑥, 𝑦 be numbers (𝑥, 𝑦 ≠ 0). Simplify each of the following expressions of numbers. 1. 𝑦 12 𝑦 12 = 2. 915 ∙ 1 915 = 3. (7(123456.789)4 )0 = 4. 22 ∙ 5. 1 25 ∙ 25 ∙ 𝑥 41 𝑦 15 ∙ 𝑦 15 𝑥 41 1 22 = = Write each number below in complete expanded exponential form. 6. 3,482 7. 65.93 8. .3875 9. 1,045,300 Lesson 6: Negative and Zero Exponents 17