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Slide 1 / 139 Slide 2 / 139 Scientific Notation 8th Grade 2015-01-15 www.njctl.org Slide 3 / 139 Slide 4 / 139 Table of Contents Links to PARCC sample questions Click on the topic to go to that section · The purpose of scientific notation · How to write numbers in scientific notation · How to convert between scientific notation and standard form Non-Calculator 5 Calculator 13 · Magnitude · Comparing numbers in scientific notation · Multiply and Divide with scientific notation · Addition and Subtraction with scientific notation · Glossary Slide 5 / 139 Vocabulary words are identified with a dotted underline. Slide 6 / 139 The charts have 4 parts. 1 Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. (Click on the dotted underline.) A whole number that can divide into another number with no remainder. 15 How many fifths are in 1 whole? The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall. 2 Its meaning How many thirds are in 1 whole? How many ninths are in 1 whole? Factor Vocab Word 3 Examples/ Counterexamples (As it is used in the lesson.) 5 R.1 3 16 5 3 is a factor of 15 3 A whole number that multiplies with another number to make a third number. 3 x 5 = 15 3 is not a factor of 16 Back to 3 and 5 are factors of 15 4 Instruction Link to return to the instructional page. Slide 7 / 139 Slide 8 / 139 Can you match these BIG objects to their weights? Purpose of Scientific Notation The Great Pyramid at Giza The Earth 300,000,000,000 kg Scientists are often confronted with numbers that look like this: 2,000,000,000,000,000, 300,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 kg 000 kg Blue Whale - largest animal on earth 600,000,000 kg Can you guess what weighs this much? 60,000,000,000,000, 000,000,000,000 kg Return to Table of Contents Slide 9 / 139 The Sun Total Human Population 180,000 kg Slide 10 / 139 Can you match these BIG objects to their weights? Can you match these small objects to their weights? grain of sand 600,000,000 kg 60,000,000,000,000, 000,000,000,000 kg Click object to reveal answer 0.00015 kg Blue Whale - largest animal on earth 180,000 kg molecule 0.000000000000000000000000030 kg steam 2,000,000,000,000,000, 000,000,000,000,000 kg 300,000,000,000 kg Slide 11 / 139 0.00000000035 kg Slide 12 / 139 Click to reveal answers. grain of sand Scientific Notation The examples were written in "standard form", the form we normally use. But the standard form is difficult to work with when a number is HUGE or tiny, if it has a lot of zeros. 0.00000000035 kg molecule Scientists have come up with a more convenient method to write very LARGE and very small numbers. 0.000000000000000000000000030 kg steam 0.00015 kg Writing numbers in scientific notation doesn't change the value of the number. Slide 13 / 139 Slide 14 / 139 Powers of Ten Scientific Notation 101 = 10 102 = 10 x 10 = 100 Scientific Notation uses Powers of 10 to write big or small numbers more conveniently. 103 = 10 x 10 x 10 = 1,000 104 = 10 x 10 x 10 x 10 = 10,000 Using scientific notation requires us to use the rules of exponents we learned earlier. While we developed those rules for all bases, scientific notation only uses base 10. 105 = 10 x 10 x 10 x 10 x 10 = 100,000 click here to see a video on powers of ten which puts our universe into perspective! click here to move from the Milky Way through space towards Earth to a oak tree, and then within a cell ! Slide 15 / 139 Slide 16 / 139 Exponent Rules Powers of Integers Remember that when multiplying numbers with exponents, if the bases are the same, you write the base and add the exponents. Powers are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition. 25 x 26 = 2(5+6) = 211 These are all equivalent: 33 x 37 = 3 103 (10)(10)(10) 1000 (3+7) = 310 108 x 10-3 = 10(8+-3) = 105 In this case, the base is 10 and the exponent is 3. 47 x 4-7 = 4(7+-7) = 40 = 1 Slide 17 / 139 1 102 x 104 = Slide 18 / 139 2 1014 x 10-6 = A 106 A 106 B 108 B 108 C 10 10 C 1010 D 1012 D 1012 Slide 19 / 139 3 Slide 20 / 139 4 10-4 x 10-6 = 104 x 10 6 = A 10-6 A 106 B 10 B 108 C 10-10 C 1010 D 10-12 D 1012 -8 Slide 21 / 139 Slide 22 / 139 Writing Large Numbers in Scientific Notation Writing Numbers in Scientific Notation Return to Table of Contents Slide 23 / 139 Scientific Notation Here are some different ways of writing 6,500. 6,500 = 6.5 thousand 6.5 thousand = 6.5 x 1,000 6.5 x 1,000 = 6.5 x 103 which means that 6,500 = 6.5 x 103 6,500 is standard form of the number and 6.5 x 103 is scientific notation These are two ways of writing the same number. Slide 24 / 139 Scientific Notation 6.5 x 103 isn't a lot more convenient than 6,500. But let's do the same thing with 7,400,000,000 which is equal to 7.4 billion which is 7.4 x 1,000,000,000 which is 7.4 x 109 Besides being shorter than 7,400,000,000 it is a lot easier to keep track of the zeros in scientific notation. And we'll see that the math gets a lot easier as well. Slide 25 / 139 Slide 26 / 139 Scientific Notation Scientific notation expresses numbers as the product of: a coefficientand 10 raised tosome power. 3.78x 106 The coefficient is always greater than or equal to one, and less than 10. In this case, the number 3,780,000 is expressed in scientific notation. Express 870,000 in scientific notation 1. Write the number without the comma. 870000 2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1. 870000 x 10 . 3. Count how many places you had to move the decimal point. This becomes the exponent of 10. 870000 x 10 . 4. Drop the zeros to the right of the right-most non-zero digit. 8.7 x 105 Slide 27 / 139 Express 53,600 in scientific notation 2 1 Express 284,000,000 in scientific notation 1. Write the number without the comma. 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1. 3. Count how many places you had to move the decimal point. This becomes the exponent of 10. 3. Count how many places you had to move the decimal point. This becomes the exponent of 10. 4. Drop the zeros to the right of the right-most non-zero digit. 4. Drop the zeros to the right of the right-most non-zero digit. Which is the correct coefficient of 147,000 when it is written in scientific notation? 4 3 Slide 28 / 139 Slide 29 / 139 5 5 Slide 30 / 139 6 Which is the correct coefficient of 23,400,000 when it is written in scientific notation? A 147 A .234 B 14.7 B 2.34 C 1.47 C 234. D .147 D 23.4 Slide 31 / 139 7 How many places do you need to move the decimal point to change 190,000 to 1.9? Slide 32 / 139 8 A 3 How many places do you need to move the decimal point to change 765,200,000,000 to 7.652? A 11 B 4 B 10 C 5 C 9 D 6 D 8 Slide 33 / 139 9 Which of the following is 345,000,000 in scientific notation? A 3.45 x 10 8 B 3.45 x 10 6 Slide 34 / 139 10 Which of these numbers, written in scientific notation, is not a number greater than one ? A .34 x 10 8 B 7.2 x 10 3 C 8.9 x 10 4 C 345 x 10 6 D 2.2 x 10 -1 D .345 x 10 9 E 11.4 x 1012 F .41 x 10 3 Slide 35 / 139 Slide 36 / 139 The mass of the solar system 300,000,000,000,000, 000,000,000,000,000, 000,000,000,000,000, 000,000,000 kg (How do you even say that number?) More Practice Slide 37 / 139 Express 9,040,000,000 in scientific notation Slide 38 / 139 Express 13,030,000 in scientific notation 1. Write the number without the comma. 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1. 3. Count how many places you had to move the decimal point. This becomes the exponent of 10. 3. Count how many places you had to move the decimal point. This becomes the exponent of 10. 4. Drop the zeros to the right of the right-most non-zero digit. 4. Drop the zeros to the right of the right-most non-zero digit. Slide 39 / 139 Express 1,000,000,000 in scientific notation 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1. 3. Count how many places you had to move the decimal point. This becomes the exponent of 10. Slide 40 / 139 11 Which of the following is 12,300,000 in scientific notation? A .123 x 10 8 B 1.23 x 10 5 C 123 x 10 5 D 1.23 x 10 7 4. Drop the zeros to the right of the right-most non-zero digit. Slide 41 / 139 Slide 42 / 139 Express 0.0043 in scientific notation 1. Write the number without the decimal point. Writing Small Numbers in Scientific Notation 2. Place the decimal so that the first number is 1 or more, but less than 10. 3. Count how many places you had to move the decimal point. The negative of this number becomes the exponent of 10. 4. Drop the zeros to the left of the left-most nonzero digit. 0043 ? 0043 . x 10 ? 0043 . x 10 1 2 3 4.3 x 10-3 Slide 43 / 139 Slide 44 / 139 Express 0.0073 in scientific notation Express 0.00000832 in scientific notation 1. Write the number without the decimal point. 1. Write the number without the decimal point. 2. Place the decimal so that the first number is 1 or more, but less than 10. 2. Place the decimal so that the first number is 1 or more, but less than 10. 3. Count how many places you had to move the decimal point. The negative of this numbers becomes the exponent of 10. 3. Count how many places you had to move the decimal point. The negative of this numbers becomes the exponent of 10. 4. Drop the zeros to the left of the left-most nonzero digit. 4. Drop the zeros to the left of the left-most nonzero digit. Slide 45 / 139 Slide 46 / 139 Scientific Notation: The Difference Between Positive & Negative Exponents 12 A 832 As you get further and further down a number line in the positive direction, your numbers are getting bigger. Therefore, really big numbers will have a positive exponent when written in scientific notation. B 83.2 C .832 D 8.32 0 20 40 60 80 100 120 140 160 180 200 As you get further and further down a number line in the negative direction, your numbers are getting smaller. Therefore, really small numbers will have a negative exponent when written in scientific notation. -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 Which is the correct decimal placement to convert 0.000832 to scientific notation? 0 Slide 47 / 139 13 Which is the correct decimal placement to convert 0.000000376 to scientific notation? Slide 48 / 139 14 How many times do you need to move the decimal point to change 0.00658 to 6.58? A 3.76 A 2 B 0.376 B 3 C 376. C 4 D 37.6 D 5 Slide 49 / 139 15 How many times do you need to move the decimal point to change 0.000003242 to 3.242? 16 Write 0.00278 in scientific notation. A 27.8 x 10 -4 A 5 B 2.78 x 10 3 B 6 C 2.78 x 10 -3 C 7 D 278 x 10 -3 D 8 Slide 51 / 139 17 Slide 50 / 139 Slide 52 / 139 Which of these numbers, written in scientific notation, is a number greater than one? A .34 x 10 -8 More Practice B 7.2 x 10 -3 C 8.9 x 10 4 D 2.2 x 10 -1 E 11.4 x 10-12 F .41 x 10 -3 Slide 53 / 139 Express 0.001003 in scientific notation Slide 54 / 139 Express 0.000902 in scientific notation 1. Write the number without the decimal point. 1. Write the number without the decimal point. 2. Place the decimal so that the first number is 1 or more, but less than 10. 2. Place the decimal so that the first number is 1 or more, but less than 10. 3. Count how many places you had to move the decimal point. The negative of this numbers becomes the exponent of 10. 3. Count how many places you had to move the decimal point. The negative of this numbers becomes the exponent of 10. 4. Drop the zeros to the left of the left-most nonzero digit. 4. Drop the zeros to the left of the left-most nonzero digit. Slide 55 / 139 Slide 56 / 139 Express 0.0000012 in scientific notation 18 Write 0.000847 in scientific notation. 1. Write the number without the decimal point. A 8.47 x 10 4 2. Place the decimal so that the first number is 1 or more, but less than 10. B 847 x 10 -4 C 8.47 x 10 -4 D 84.7 x 10 -5 3. Count how many places you had to move the decimal point. The negative of this numbers becomes the exponent of 10. 4. Drop the zeros to the left of the left-most nonzero digit. Slide 57 / 139 Slide 58 / 139 Express 3.5 x 10 4 in standard form Converting to Standard Form Return to Table of Contents 1. Write the coefficient. 3.5 2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative. 3.50000 3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. 35000.0 4. Drop unnecessary zeros and add comma, as necessary. 35,000 Slide 59 / 139 Express 1.02 x 10 in standard form 6 Slide 60 / 139 Express 3.42 x 10 in standard form -3 1. Write the coefficient. 1. Write the coefficient. 2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative. 2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative. 3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. 3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. 4. Drop unnecessary zeros and add comma, as necessary. 4. Drop unnecessary zeros and add comma, as necessary. Slide 61 / 139 Express 2.95 x 10 in standard form -4 Slide 62 / 139 19 1. Write the coefficient. 2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative. How many times do you need to move the decimal and which direction to change 7.41 x 10 -6 into standard form? A 6 to the right B 6 to the left 3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. C 7 to the right D 7 to the left 4. Drop unnecessary zeros and add comma, as necessary. Slide 63 / 139 20 How many times do you need to move the decimal and which direction to change 4.5 x 10 10 into standard form? Slide 64 / 139 21 Write 6.46 x 10 4 in standard form. A 646,000 B 0.00000646 A 10 to the right C 64,600 B 10 to the left D 0.0000646 C 11 to the right D 11 to the left Slide 65 / 139 22 Write 3.4 x 10 3 in standard form. Slide 66 / 139 23 Write 6.46 x 10 -5 in standard form. A 3,400 A 646,000 B 340 B 0.00000646 C 34,000 C 0.00646 D 0.0034 D 0.0000646 Slide 67 / 139 24 Write 1.25 x 10 -4 in standard form. 25 Write 4.56 x 10 -2 in standard form. A 125 A 456 B 0.000125 B 4560 C 0.00000125 C 0.00456 D 4.125 D 0.0456 Slide 69 / 139 26 Slide 68 / 139 Write 1.01 x 10 9 in standard form. A 101,000,000,000 B 1,010,000,000 Slide 70 / 139 Using a Calculator for Scientific Notation When entering numbers into a calculator that are in scientific notation, you can use the EE button. It means "x 10 to the power of". C 0.00000000101 D 0.000000101 This button eliminates the "x 10" of a number in scientific notation. So 9 x 108 is entered into the calculator using 9 EE 8 and shows up at the top as 9E8. Slide 71 / 139 Using a Calculator for Scientific Notation Slide 72 / 139 Using a Calculator for Scientific Notation Enter the following numbers into the calculator using the "EE" button to determine its value in standard form. a) 4 x 102 b) 5.7 x 10-3 c) 9.87 x 104 d) 1.43 x 10-1 When reading a calculator that has a number in scientific notation, remember that the "E" stands for "x 10 to the power of". Which number written in standard form represents the number in the calculator below? 3.2E9 Slide 73 / 139 Using a Calculator for Scientific Notation When reading a calculator that has a number in scientific notation, remember that the "E" stands for "x 10 to the power of". 4.21E-11 Slide 74 / 139 27 Which number written in standard form represents the number in the calculator below? A 0.000000000482 B 0.0000000000482 C 4,820,000,000,000 Which number written in standard form represents the number in the calculator below? D 48,200,000,000 Slide 75 / 139 28 Which number written in standard form represents the number in the calculator below? A 0.000000653 B 0.00000653 Slide 76 / 139 29 Which number written in standard form represents the number in the calculator below? A 0.000000000974 6.53E-6 B 0.0000000000974 C 6,530,000 C 9,740,000,000,000 D 653,000,000 D 97,400,000,000 Slide 77 / 139 30 Which number written in standard form represents the number in the calculator below? A 0.00000407 B 0.000000407 4.82E10 Slide 78 / 139 31 Liz saw this number on her calculator screen. Which numbers represent the number Liz saw? A 0.0000006 4.07E6 9.74E-10 B 0.00000006 C 4,070,000 C -6,000,000 D 407,000,000 D -60,000,000 From PARCC sample test Slide 79 / 139 Slide 80 / 139 Magnitude Scientific notation always uses decimal notation that is bigger than 1 but smaller than 10. Why? Magnitude This is due to magnitude. Magnitude is how we can observe very large or very small numbers and easily compare them. The magnitude of a number is the exponent when the number is written in scientific notation. Below are a few examples. 8304 = 8.304 x 103 - the order of magnitude is 3 Return to Table of Contents 20,000 = 2 x 104 – the order of magnitude is 4 0.000034 = 3.4 x 10-5 – the order of magnitude is -5 Slide 81 / 139 Slide 82 / 139 Write each of the following in Scientific Notation first and then indicate the order of magnitude. Scientific Notation Order of Magnitude 6214 Application Let J represent the world population in 1950. J = 2,556,000,053. Find the smallest power of 10 that will exceed J. 472.17 The number above (J) has 10 digits and is smaller than a whole number with 11 digits. 813000000 .000253 (10,000,000,000 or 1010 therefore J<1010) .00647 The answer is 10. .00000049 Slide 83 / 139 Application Let K represent the national debt in 1950. Slide 84 / 139 32 If m = 149,162,536,496,481,100 find the smallest power of 10 that will exceed m. K = 257,357,352,351. Find the smallest power of 10 that will exceed K. ( (Derived from Slide 85 / 139 Slide 86 / 139 33 What is the smallest power of 10 that will exceed 5,321? (Derived from find the smallest power of 10 that will ( ( (Derived from 34 If m = 628 exceed m? Slide 87 / 139 Slide 88 / 139 35 What would the negative exponent be used to express the number ? 36 The chance of a shark bite is and the chance of dying from a snake bite is which are more likely to be bit by? A both a the same B the snake C the shark D neither ( (Derived from Slide 89 / 139 ( (Derived from Slide 90 / 139 Click for web site Comparing Numbers Written in Scientific Notation Return to Table of Contents Slide 91 / 139 Comparing numbers in scientific notation When the exponents are different, just compare the exponents. Answer Comparing numbers in scientific notation Slide 92 / 139 First, compare the exponents. Whichever number has the larger exponent is the larger number. = < If the exponents are different, the coefficients don't matter; they have a smaller effect. > 9.99 x 10 3 2.17 x 10 4 1.02 x 10 2 8.54 x 10 -3 6.83 x 10 -9 3.93 x 10 -2 just drag the sign that is correct Slide 93 / 139 Slide 94 / 139 Comparing numbers in scientific notation Comparing numbers in scientific notation If the exponents are the same, compare the coefficients. When the exponents are the same, just compare the The larger the coefficient, the larger the number (if the exponents are the same). > 5.67 x 103 4.67 x 103 4.32 x 106 4.67 x 106 2.32 x 1010 3.23 x 1010 Slide 95 / 139 37 Which is ordered from least to greatest? = < coefficients. Slide 96 / 139 38 Which is ordered from least to greatest? A I, II, III, IV I. 1.0 x 10 5 A I, II, III, IV I. 1.0 x 102 B IV, III, I, II II. 7.5 x 10 6 B IV, III, I, II II. 7.5 x 10 6 C I, IV, II, III III. 8.3 x 10 4 C I, IV, II, III III. 8.3 x 10 9 D III, I, II, IV IV. 5.4 x 107 D I, II, IV, III IV. 5.4 x 107 Slide 97 / 139 39 Which is ordered from least to greatest? Slide 98 / 139 40 Which is ordered from least to greatest? A I, II, III, IV I. 1 x 102 A II, III, I, IV I. 1 x 10-2 B IV, III, I, II II. 7.5 x 103 B IV, III, I, II II. 7.5 x 10-24 C III, IV, II, I III. 8.3 x 10-2 C III, IV, II, I III. 8.3 x 10-15 D III, IV, I, II IV. 5.4 x 10-3 D III, IV, I, II Slide 99 / 139 41 Which is ordered from least to greatest? IV. 5.4 x 102 Slide 100 / 139 42 Which is ordered from least to greatest? A I, II, III, IV I. 1.0 x 102 A I, II, III, IV I. 1.0 x 106 B IV, III, I, II II. 7.5 x 102 B IV, III, I, II II. 7.5 x 106 C I, IV, II, III III. 8.3 x 102 C I, IV, II, III III. 8.3 x 106 D III, IV, I, II IV. 5.4 x 102 D III, IV, I, II Slide 101 / 139 43 Which is ordered from least to greatest? IV. 5.4 x 107 Slide 102 / 139 44 Which is ordered from least to greatest? A I, II, III, IV I. 1.0 x 103 A I, II, III, IV I. 2.5 x 10-3 B IV, III, I, II II. 5.0 x 103 B IV, III, I, II II. 5.0 x 10-3 C I, IV, II, III III. 8.3 x 106 C I, IV, II, III III. 9.2 x 10-6 D III, IV, I, II IV. 9.5 x 106 D III, IV, I, II IV. 4.2 x 10-6 Slide 103 / 139 Slide 104 / 139 Multiplying Numbers in Scientific Notation Multiplying Numbers in Scientific Notation Evaluate: (6.0 x 104 )(2.5 x 102 ) Multiplying with scientific notation requires at least three (and sometimes four) steps. 1. Multiply the coefficients 2. Multiply the powers of ten applying the rule of exponents 3. Combine those results 4. Put in proper form 1. Multiply the coefficients 6.0 x 2.5 = 15 2. Multiply the powers of ten applying the rule of exponents 104 x 102 = 106 3. Combine those results 15 x 106 4. Put in proper form 1.5 x 107 Return to Table of Contents Slide 105 / 139 Multiplying Numbers in Scientific Notation Evaluate: (4.80 x 106 )(9.0 x 10-8 ) 1. Multiply the coefficients 2. Multiply the powers of ten applying the rule of exponents 3. Combine those results Slide 106 / 139 45 Evaluate (2.0 x 10 -4)(4.0 x 107). Express the result in scientific notation. A 8.0 x 10 11 B 8.0 x 103 C 5.0 x 103 D 5.0 x 10 11 E 7.68 x 10-28 F 7.68 x 10-28 4. Put in proper form Slide 107 / 139 46 Evaluate (5.0 x 10 6 )(7.0 x 10 7 ) Slide 108 / 139 47 Evaluate (6.0 x 10 2 )(2.0 x 10 3 ) A 3.5 x 10 13 A 1.2 x 10 6 B 3.5 x 10 B 1.2 x 10 1 C 3.5 x 10 1 C 1.2 x 10 5 D 3.5 x 10 -1 D 3.0 x 10 -1 E 7.1 x 10 13 E 3.0 x 10 5 F 7.1 x 10 1 F 3.0 x 10 1 14 Slide 109 / 139 48 Evaluate (1.2 x 10 -6 )(2.5 x 10 3 ). Express the result in scientific notation. Slide 110 / 139 49 Evaluate (1.1 x 104)(3.4 x 106). Express the result in scientific notation. A 3 x 10 3 B 3 x 10 -3 A 3.74 x 1024 B 3.74 x 1010 C 30 x 10 -3 C 4.5 x 1024 D 0.3 x 10 -18 D 4.5 x 1010 E 30 x 10 E 37.4 x 1024 18 Slide 111 / 139 50 Evaluate (3.3 x 10 4 )(9.6 x 10 3 ). Express the result in scientific notation. A 31.68 x 10 7 B 3.168 x 10 C 3.2 x 10 Slide 112 / 139 51 Evaluate (2.2 x 10 -5 )(4.6 x 10 -4 ). Express the result in scientific notation. A 10.12 x 10 -20 8 B 10.12 x 10 -9 C 1.012 x 10 -10 7 D 32 x 10 8 D 1.012 x 10 -9 30 x 10 7 E 1.012 x 10 -8 E Slide 113 / 139 Dividing Numbers in Scientific Notation Dividing with scientific notation follows the same basic rules as multiplying. 1. Divide the coefficients 2. Divide the powers of ten applying the rule of exponents 3. Combine those results 4. Put in proper form Slide 114 / 139 Division with Scientific Notation Evaluate: 5.4 x 106 9.0 x 102 1. Divide the coefficients 5.4 ÷ 9.0 = 0.6 2. Divide the powers of ten applying the rule of exponents 106 ÷ 102 = 104 3. Combine those results 0.6 x 104 4. Put in proper form 6.0 x 103 Slide 115 / 139 Division with Scientific Notation Evaluate: 4.4 x 106 Slide 116 / 139 52 Evaluate 4.16 x 10 5.2 x 10 Express the result in scientific notation. -9 -5 1.1 x 10 -3 A 0.8 x 10 -4 1. Divide the coefficients 2. Divide the powers of ten applying the rule of exponents B 0.8 x 10 -14 C 0.8 x 10 -5 3. Combine those results D 8 x 10 -4 4. Put in proper form E 8 x 10 -5 Slide 117 / 139 53 Evaluate 7.6 x 10 4 x 10 Express the result in scientific notation. -2 -4 Slide 118 / 139 54 Evaluate 8.2 x 10 2 x 10 Express the result in scientific notation. A 1.9 x 10 -2 A 4.1 x 10 -10 B 1.9 x 10 -6 B 4.1 x 10 4 C 1.9 x 10 2 C 4.1 x 10 -4 D 1.9 x 10 -8 D 4.1 x 10 21 1.9 x 10 8 E 4.1 x 10 10 E Slide 119 / 139 55 Evaluate 3.2 x 10 6.4 x 10 Express the result in scientific notation. -2 -4 3 7 Slide 120 / 139 56 The point on a pin has a diameter of approximately 1 x 10 meters. If an atom has a diameter of 2 x 10 meters, about how many atoms could fit across the diameter of the point of a pin? -4 -10 A .5 x 10 -6 B .5 x 10 -2 A 50,000 C .5 x 10 2 B 500,000 D 5 x 10 1 C 2,000,000 E D 5,000,000 5 x 10 3 Question from ADP Algebra I End-of-Course Practice Test Slide 121 / 139 Slide 122 / 139 57 The body of a 154 pound person contains approximately 2 X 10-1 milligrams of gold and 6 X 101 milligrams of aluminum. Based on this information, the number of milligrams of aluminum in the body is how many times the number of milligrams of gold in the body? Addition and Subtraction with Scientific Notation Numbers in scientific notation can only be added or subtracted if they have the same exponents. If needed, an intermediary step is to rewrite one of the numbers so it has the same exponent as the other. Return to Table of Contents From PARCC sample test Slide 123 / 139 Slide 124 / 139 Addition and Subtraction This problem is slightly more difficult because you need to add one extra step at the end. This is the simplest example of addition 4.0 x 10 + 5.3 x 10 = 3 Addition and Subtraction 3 8.0 x 10 + 5.3 x 10 = 3 Since the exponents are the same (3), just add the coefficients. 4.0 x 10 + 5.3 x 10 = 9.3 x 10 3 3 3 Since the exponents are the same (3), just add the coefficients. This just says 8.0 x 10 + 5.3 x 10 = 13.3 x 10 3 4.0 thousand + 5.3 thousand 9.3 thousand. 8.0 x 104 + 5.3 x 103 = This requires an extra step at the beginning because the exponents are different. We have to either convert the first number to 80 x 103 or the second one to 0.53 x 104 . The latter approach saves us a step at the end. 8.0 x 104 + 0.53 x 104 = 8.53 x 104 Once both numbers had the same exponents, we just add the coefficient. Note that when we made the exponent 1 bigger, that's makes the number 10x bigger; we had to make the coefficient 1/10 as large to keep the number the same. 3 3 But that is not proper form, since 13.3 > 10; it should be written as 1.33 x 10 4 Slide 125 / 139 Addition and Subtraction 3 Slide 126 / 139 58 The sum of 5.6 x 10 3 and 2.4 x 10 3 is A 8.0 x 10 3 B 8.0 x 10 6 C 8.0 x 10 -3 D 8.53 x 10 3 Slide 127 / 139 Slide 128 / 139 59 8.0 x 103 minus 2.0 x 103 is 60 7.0 x 103 plus 2.0 x 102 is A 6.0 x 10-3 A 9.0 x 103 B 0 6.0 x 10 B 9.0 x 105 C 6.0 x 103 C 7.2 x 103 D 7.8 x 103 D 7.2 x 102 Slide 129 / 139 Slide 130 / 139 61 3.5 x 105 plus 7.8 x 105 is A 11.3 x 105 B 1.13 x 104 C 1.13 x 106 D 11.3 x 1010 Glossary Return to Table of Contents Slide 131 / 139 Slide 132 / 139 Base Coefficient The number that is going to be raised to a power. It is multiplied the number of times shown in the power. 188 335 7 3 215 29 112 4 A number used to multiply a variable. In scientific notation, ten to the power of 3 103 = the base will always = 10 10 x 10 x 10 = 1,000 Back to Instruction 3y 19z 6.5 x 103 A factor of a term. scientific notation: .000000459 a coefficient and 10 raised to some power 3.78 x 106 4.59 x 10-7 Back to Instruction Slide 133 / 139 Slide 134 / 139 Power Blue Whale A convenient system scientists developed to rewrite big or small numbers using powers of 10 that does not change the value. A number that shows you A quick way to write how many times to use the repeated number in a multiplication. multiplication. a.k.a. scientific notation: Exponent big numbers a coefficient 180,000 kg = and 10 raised to or ten to the power of 3 Index 103 = some power 1.8 x 105 3.78 x 106 10 x 10 x 10 = 1,000 Scientific Notation Back to small numbers a coefficient and 10 raised to some power. 0.00015 kg = 3.78 x 106 1.5 x 10-4 Instruction Slide 135 / 139 Back to Instruction Slide 136 / 139 Standard Form A number whose scientific form has been expanded. 4,500,000 0.00000032 0.006789 120,000 The most familiar form of a number. Standard Form: 6,500 vs. Scientific Form: 6.5 x 103 *Note* this is not the "correct" form but the most recognizable form. Back to Back to Instruction Instruction Slide 137 / 139 Slide 138 / 139 Back to Back to Instruction Instruction Slide 139 / 139 Back to Instruction