ScientificNotation

ScientificNotation - Module 1 – Scientific Notation How...

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Unformatted text preview: Module 1 – Scientific Notation How to Use These Lessons 1. Read the lesson. Work the questions (Q) as you go. To work the questions, use this method. • As you turn to a new page, cover the page on the right with a sheet of paper. • As you start each new page, if you see 5 stars ( * * * * * ) on the page, cover the text below the stars. As a cover sheet, use either overlapping sticky notes or a folded sheet of paper. • In your problem notebook, answer the question (Q) above the * * * * * . Then slide down the cover sheet to the next set of * * * * * and check your answer. If you need a hint, read a part of the answer, then re-cover the answer and try the problem again. 2. Memorize the rules, then do the Practice that follows. When working questions (Q) in a lesson, you may look back at the rules, but learn the rules before starting the Practice problems. Treat Practice as a practice test. Try every other problem of a Practice set on the first day and the remaining problems in your next study session. This spacing will help you to remember new material. On both days, try to work the Practice without looking back at the rules. Step-by-step Practice answers are the end of each lesson. If you need a hint, read a part of the answer and try again. 3. How many Practice problems should you do? It depends on your background. These lessons are intended to • refresh your memory on topics you once knew, and • fill-in the gaps for topics that are less familiar. If you know a topic well, read the lesson for review, then do a few problems on each Practice set. Be sure to do the last problem (usually the most challenging). If a topic is unfamiliar, do more problems. 4. Work Practice problems at least 3 days a week. Chemistry is cumulative. What you learn early you will need later. To retain what you learn, space your study of a topic over several days. Science has found that your memory tends to retain what it uses repeatedly, but to remember for only a few days what you do not practice over several days. If you wait until a quiz deadline to study, what you learn may remain in memory for a day or two, but on later tests, it will tend to be forgotten. Begin lessons on new topics early, preferably before the topic is introduced in lecture. 5. Memorize what must be memorized. Use flashcards and other memory aids. Chemistry is not easy, but if you work at a steady pace, you will achieve success. ©2010 www.ChemReview.Net v. u1 Page 1 Module 1 – Scientific Notation If you have previously taken a course in chemistry, many topics in Modules 1 to 4 will be review. Therefore: if you can pass the pre-test for a lesson, skip the lesson. If you need a bit of review to refresh your memory, move quickly, doing the last few problems of each Practice set. On topics that are less familiar, complete more Practice. Module 1 – Scientific Notation Timing: Module 1 should be done as soon as you are assigned problems that use exponential notation. If possible, do these lessons before attempting problems in other textbooks. Additional Math Topics Powers and roots of exponential notation are covered in Lesson 28B. Complex units such as are covered in Lesson 17C. atm ● L (mole)( atm ● L ) mole ● K Those lessons may be done at any time after Module 1. Calculators and Exponential Notation To multiply 492 x 7.36, the calculator is a useful tool. However, when using exponential notation, you will make fewer mistakes if you do as much exponential math as you can without a calculator. These lessons will review the rules for doing exponential math “in your head.” The majority of problems in Module 1 will not require a calculator. Problems that require a calculator will be clearly identified. You are encouraged to try complex problems with the calculator after you have tried them without. This should help you to decide when, and when not, to use a calculator. ***** Lesson 1A: Moving the Decimal Pretest: If you get a perfect score on this pretest, skip to Lesson 1B. Otherwise, complete Lesson 1A. Answers are at the end of each lesson. Change these to scientific notation. 1. 9,400 x 103 = ___________________ 2. 0.042 x 106 = _________________ 3. ─ 0.0067 x 10―2 = _________________ 4. ─ 77 = _________________ ***** ©2010 www.ChemReview.Net v. u1 Page 2 Module 1 – Scientific Notation Powers of 10 In science, we often deal with very large and very small numbers. For example: A drop of water contains about 1,500,000,000,000,000,000,000 molecules. An atom of neon has a radius of about 0.0000000070 centimeters. Large and small numbers are more easily expressed using powers of 10. For the measurements above, we write • A drop of water contains about 1.5 x 1021 molecules. • An atom of neon gas a radius of about 7.0 x 10―9 centimeters. Below are the numbers that correspond to powers of 10. Note the relationship between the exponents and position of the decimal point in the numbers as you go down the sequence. 106 = 1,000,000 103 = 1,000 = 10 x 10 x 10 102 = 100 1 0 1 = 10 100 = 1 (Anything to the zero power equals one.) 10―1 = 0.1 10―2 = 0.01 = 1/102 = 1/100 10―3 = 0.001 To change from powers of 10 to the numbers they represent, use these steps. 1. To convert an exponential term that has a positive power of 10 to a standard number, • write 1, then after 1 add the number of zeros equal to the exponent. Example: 102 = 100 • Another way to state this rule: From 1, move the decimal to the right by the number of places in the exponent. 2. To convert a negative power of 10 to a number, • From 1, move the decimal to the left by the number of places equal to the exponent of 10 after its negative sign. Example: Practice A: 10―2 = 0 .01 ↑∪∪ Write your answers, then check them at the end of this lesson. 1. Write these as regular numbers without an exponential term. a. 104 = _______________ c. 107 = _______________ ©2010 www.ChemReview.Net v. u1 b. 10―4 = ______________ d. 10―5 = ______________ e. 100 = ________ Page 3 Module 1 – Scientific Notation Multiplying and Dividing By 10, 100, 1000 When multiplying or dividing by numbers that are positive powers of 10, such as 100 or 10,000, use the following rules. 1. When multiplying a number by a 10, 100, 1000, etc., move the decimal to the right by the number of zeros in the 10, 100, or 1000. Examples: 72 x 100 = 7200 ─ 0.0624 x 1,000 = ─ 62.4 2. When dividing a number by 10, 100, or 1000, etc., move the decimal to the left by the number of zeros in the 10, 100, or 1000. Examples: 34.6/1000 = 0.0346 0.47/100 = 0.0047 ↑∪∪∪ 3. When writing a number that has a value between ─1 and 1 (a number that “begins with a decimal point”), always place a zero in front of the decimal point. Example: Do not write .42 or ─ .74 ; do write 0.42 or ─ 0.74 During your written calculations, the zero in front helps in seeing your decimals. Practice B: Write your answers, then check them at the end of this lesson. 1. When dividing by 1,000 move the decimal to the _______________ by _____ places. 2. Write these answers as regular numbers. a. 0.42 x 1000 = b. 63/100 = c. ─ 74.6/10,000 = Numbers in Exponential Notation Numbers such as 123 or 0.24 are said to be written in fixed decimal or fixed notation. Values in exponential notation are written as a number times 10 to a whole-number power. Examples: 5,250 = 5.25 x 103 ─ 0.00065 = ─ 6.5 x 10―4 Numeric values represented in exponential notation can be described as having three parts. In ─ 6.5 x 10―4, • The ─ in front is the sign. • the 6.5 is termed the significand or digit or mantissa or coefficient. • The 10―4 is the exponential term: the base is 10 and the exponent (or power) is ―4. In these lessons we will refer to the two parts of exponential notation after the sign as the significand and exponential term. ↓ ─ 6.5 x 10―4 ↑ ↑ significand exponential sign You should also learn (and use) any alternate terminology preferred in your course. ©2010 www.ChemReview.Net v. u1 Page 4 Module 1 – Scientific Notation Converting Exponential Notation to Numbers To convert from exponential notation to a number without an exponential term, use the following rules. a. The sign in front never changes. The sign is independent of the terms after the sign. b. If the significand is multiplied by a • positive power of 10, move the decimal point in the significand to the right by the same number of places as the value of the exponent; Examples: • 2 x 102 = 2 00 ∪∪↑ ─ 0 .0033 x 103 = ∪∪∪↑ ─ 3.3 negative power of 10, move the decimal point in the significand to the left by the same number of places as the number after the minus sign of the exponent. Examples: 2 x 10―2 = 0 .02 ↑∪∪ ─ 7,653 x 10―3 = ─ 7,653 x 0.001 = ─ 7 .653 ↑∪∪∪ Practice C: Convert these to fixed decimal notation. 1. 3 x 103 = _____________________ 2. 5.5 x 10―4 = _________________________ 3. 0.77 x 106 = __________________ 4. ─ 95 x 10―4 = ______________________ Changing Exponential to Scientific Notation In chemistry, it is often required that numbers that are very large or very small be written in scientific notation (also called standard exponential notation). We do this because values written in scientific notation are easier to compare: there are many equivalent ways to write a given number in exponential notation, but only one correct way to write the number in scientific notation. Scientific notation is a special case of exponential notation in which the significand is 1 or greater, but less than 10, and is multiplied by 10 to a whole-number power. Another way to say this: in scientific notation, the decimal point in the significand must be after the first digit that is not a zero. Example: ─ 0.057x 10―2 in scientific notation is written as ─ 5.7 x 10―4 . The decimal must be moved to after the first number that is not a zero: the 5. To convert from exponential to scientific notation, • move the decimal in the significand to after the first digit that is not a zero, then • adjust the exponent to keep the same numeric value. ©2010 www.ChemReview.Net v. u1 Page 5 Module 1 – Scientific Notation When moving a decimal point, use these rules. 1. When moving the decimal, the sign in front never changes. 2. If moving the decimal Y times to make the significand larger, make the power of 10 smaller by a count of Y. Example. Converting exponential to scientific notation: 0.045 x 10 5 = 4 . 5 x 10 3 ∪∪↑ To convert to scientific notation, the decimal must be after the 4. Move the decimal two times to the right. This makes the significand 100 times larger. To keep the same numeric value, lower the exponent by 2, making the 10x value 100 times smaller. 3. When moving the decimal Y times to make the significand smaller, make the power of 10 larger by a count of Y. Example. Convert to scientific notation: ─ 8 , 544 x 10 ― 7 = ─ 8 . 544 x 10 ― 4 ↑∪∪∪ To convert to scientific notation, you must move the decimal 3 places to the left. This makes the significand 1,000 times smaller. To keep the same numeric value, increase the exponent by 3, making the 10x value 1,000 times larger. Remember, 10─4 is 1,000 times larger than 10─7. It helps to recite, every time you move a decimal, for the terms after the sign in front: “If one gets smaller, the other gets larger. If one gets larger, the other gets smaller.” Practice D: Change these to scientific notation. Check answers at the end of the lesson. 1. 5,420 x 103 = 2. 0.0067 x 10― 4 = ___________________ 3. 0.020 x 103 = 4. ─ 870 x 10― 4 = _____________________ 5. 0.00492 x 10― 12 = 6. ─ 602 x 1021 = _____________________ Converting Numbers to Scientific Notation To convert regular (fixed decimal) numbers to exponential notation, we will need these rules. • Any number to the zero power equals one. 20 = 1. • 420 = 1. Exponential notation most often uses 100 = 1. Since any number can be multiplied by one without changing its value, any number can be multiplied by 100 without changing its value. Example: 42 = 42 x 1 = 42 x 100 in exponential notation = 4.2 x 101 in scientific notation. ©2010 www.ChemReview.Net v. u1 Page 6 Module 1 – Scientific Notation To convert fixed notation to scientific notation, the steps are 1. Add x 100 after the number. 2. Apply the rules that convert exponential to scientific notation: • The sign in front does not change. • Write the decimal after the first digit that is not a zero. • Adjust the power of 10 to compensate for moving the decimal. Try: Q. Using the steps above, convert these numbers to scientific notation. a. ***** 943 b. ─ 0.00036 (See How To Use These Lessons, Point 1, on page 1). Answers: 943 = 943 x 100 = 9.43 x 102 in scientific notation. ─ 0.00036 = ─ 0.00036 x 100 = ─ 3.6 x 10―4 in scientific notation. When a number is converted to scientific notation, numbers after the sign in front that are • larger than one have positive exponents (zero and above) in scientific notation; • between zero and one have negative exponents in scientific notation; and • the number of places that the decimal in a number moves is the number after the sign in its exponent. Note how these three rules apply to the two answers above. Note also that in both exponential and scientific notation, whether the sign in front is positive or negative has no impact on the sign of the exponential term. The exponential term indicates the position of the decimal point, and not whether a value is positive or negative. Practice E 1. Which lettered parts in Problem 2 below must have exponentials that are negative when written in scientific notation? 2. Change these to scientific notation. a. 6,280 = b. 0.0093 = _________________________ c. 0.741 = _____________________ d. ─ 1,280,000 = _____________________ 3. Complete the problems in the pretest at the beginning of this lesson. Study Summary In your problem notebook, write a list of rules in this lesson that were unfamiliar, needed reinforcement, or you found helpful. Then condense the wording, number the points, and write and recite until you can write them from memory. Then complete the problems below. ©2010 www.ChemReview.Net v. u1 Page 7 Module 1 – Scientific Notation Practice F Check and do every other letter. If you miss one, do another letter for that set. Save a few parts for your next study session. 1. Write these answers in fixed decimal notation. a. 924/10,000 = b. 24.3 x 1000 = c. ─ 0.024/10 = 2. Convert to scientific notation. a. 0.55 x 105 c. 940 x 10―6 d. 0.00032 x 101 c. 0.023 b. 0.0092 x 100 d. 0.00067 3. Write these numbers in scientific notation. a. 7,700 ANSWERS Pretest: b. 160,000,000 (To make answer pages easy to locate, use a sticky note.) 1. 9.4 x 106 2. 4.2 x 104 Practice A: 1a. 10,000 b. 0.0001 3. ─ 6.7 x 10―5 c. 10,000,000 4. ─ 7.7 x 101 d. 0.00001. e. 1 Practice B: 1. When dividing by 1,000 , move the decimal to the left by 3 places. 2a. 0.42 x 1000 = 420 2b. 63/100 = 0.63 (must have zero in front) Practice C: 1. 3,000 2. 0.00055 3. 770,000 c. ─ 74.6/10,000 = ─ 0.00746 4. ─ 0.0095 Practice D: 1. 5.42 x 106 2. 6.7 x 10―7 3. 2.0 x 101 4. ─8.7 x 10―2 5. 4.92 x 10―15 6. ─6.02 x 1023 Practice E: 1. 2b and 2c Practice F: 1a. 0.0924 2a. 6.28 x 103 2b. 9.3 x 10―3 1b. 24,300 2a. 5.5 x 104 2b. 9.2 x 10―1 3a. 7.7 x 103 ***** 3b. 1.6 x 108 2c. 7.41 x 10―1 2d. ─ 1.28 x 106 1c. ─ 0.0024 2c. 9.4 x 10―4 2d. 3.2 x 10―3 3c. 2.3 x 10―2 3d. 6.7 x 10―4 Lesson 1B: Calculations Using Exponential Notation Pretest: If you can answer all of these three questions correctly, you may skip to Lesson 1C. Otherwise, complete Lesson 1B. Answers are at the end of the lesson. Do not use a calculator. Convert final answers to scientific notation. 1. (2.0 x 10―4) (6.0 x 1023) = 2. 1023 = (100)(3.0 x 10―8) 3. (─ 6.0 x 10―18) ─ (─ 2.89 x 10―16) = ***** ©2010 www.ChemReview.Net v. u1 Page 8 Module 1 – Scientific Notation Adding and Subtracting Exponential Notation To add or subtract exponential notation without a calculator, the standard rules of arithmetic can be applied – if all of the numbers have the same exponential term. Re-writing numbers to have the same exponential term usually results in values that are not in scientific notation. That’s OK. During calculations, the rule is to work in exponential notation, allowing flexibility with decimal point positions, then to convert to scientific notation at the final step. To add or subtract numbers with exponential terms, you may convert all of the exponential terms to any consistent power of 10. However, it usually simplifies the arithmetic if you convert all values to the same exponential as the largest of the exponential terms being added or subtracted. The rule is To add or subtract exponential notation by hand, make all of the exponents the same. The steps are To add or subtract exponential notation without a calculator, 1. Re-write each number so that all of the significands are multiplied by the same power of 10. Converting to the highest power of 10 being added or subtracted is suggested. 2. Write the significands and exponentials in columns: numbers under numbers (lining up the decimal points), x under x, exponentials under exponentials. 3. Add or subtract the significands using standard arithmetic, then attach the common power of 10 to the answer. 4. Convert the final answer to scientific notation. Follow how the steps are applied in this Example: ( 40.71 x 108 ) + ( 222 x 106 ) = ( 40.71 x 108 ) + ( 2.22 x 108 ) = 40.71 x 108 + 2.22 x 108 42.93 x 108 = 4.293 x 109 Using the steps and the method shown in the example, try the following problem without a calculator. In this problem, do not round numbers during or after the calculation. Q. ( 32.464 x 101 ) ─ (16.2 x 10―1 ) = ? ***** ©2010 www.ChemReview.Net v. u1 Page 9 Module 1 – Scientific Notation ( 32.464 x 101 ) ─ (16.2 x 10―1 ) = ( 32.464 x 101 ) ─ (0.162 x 10+1 ) = A. 32.464 x 101 ─ 0.162 x 101 32.302 x 101 (101 has a higher value than 10―1) = 3.2302 x 102 Let’s do problem 1 again. This time, below, convert the exponential notation to regular numbers, do the arithmetic, then convert the final answer to scientific notation. 32.464 x 101 ─ 16.2 x 10―1 = = ***** 32.464 x 101 = ─ 1 6 .2 x 10―1 = 324.64 ─ 1.62 323.02 = 3.2302 x 102 The answer is the same either way, as it must be. This “convert to regular numbers” method is an option when the exponents are close to 0. However, for exponents such as 1023 or 10―17, we will want to use the method above that includes the exponential, but adjusts so that all of the exponentials are the same. The Role of Practice Do as many Practice problems as you need to feel “quiz ready.” • If the material in a lesson is relatively easy review, do the last problem on each series of similar problems. • If the lesson is less easy, put a check by ( ) and then work every 2nd or 3rd problem. If you miss one, do some similar problem in the set. • Save a few problems for your next study session -- and quiz/test review. During Examples and Q problems, you may look back at the rules, but practice writing and recalling new rules and steps from memory before the Practice. If you use the Practice to learn the rules, it will be difficult to find time for all of the problems you will need to do. If you use Practice to apply rules that are in memory, you will need to do fewer problems to be “quiz ready.” Practice A: Try these without a calculator. On these, don’t round. Do convert final answers to scientific notation. Check your answer after each problem. 1. 2. 64.202 x 1023 + 13.2 x 1021 (61 x 10―7) + (2.25 x 10―5) + (212.0 x 10―6) = ©2010 www.ChemReview.Net v. u1 Page 10 Module 1 – Scientific Notation 3. ( ― 54 x 10―20 ) + ( ― 2.18 x 10―18 ) = 4. ( ― 21.46 x 10―17 ) ― ( ― 3,250 x 10―19 ) = Multiplying and Dividing Powers of 10 The following boxed rules should be recited until they are memorized. 1. When you multiply exponentials, you add the exponents. Examples: 2. 103 x 102 = 105 10―5 x 10―2 = 10―7 10―3 x 105 = 102 When you divide exponentials, you subtract the exponents. Examples: 103/102 = 101 10―5/102 = 10―7 10―5/10―2 = 10―3 When subtracting, remember: Minus a minus is a plus. 106―(―3) = 106+3 = 109 3. When you take the reciprocal of an exponential, change the sign. This rule is often remembered as: When you take an exponential term from the bottom to the top, change the sign. 1 = 10―3 ; 1/10―5 = 105 103 Why does this work? Rule 2: 1 = 100 103 103 Example: = 100―3 = 10―3 4. When fractions include several terms, it often helps to simplify the top and bottom separately, then divide. 10―3 Example: 105 x 10―2 = 10―3 = 103 10―6 Try the following problem. Q. Without using a calculator, simplify the top, then the bottom, then divide. 10―3 x 10―4 105 x 10―8 = = * * * * * (See How To Use These Lessons, Point 1, on page 1). Answer: 10―3 x 10―4 = 105 x 10―8 10―7 = 10―7―(―3) = 10―7+3 = 10―4 10―3 Practice B: Write answers as 10 to a power. Do not use a calculator. Check your answers at the end of the lesson. 1. 106 x 102 = ©2010 www.ChemReview.Net 2. v. u1 10―5 x 10―6 = Page 11 Module 1 – Scientific Notation 3. 10―5 = 10―4 4. 10―3 = 105 5. 1 = ―4 10 6. 1/1023 = 7. 103 x 10―5 = 10―2 x 10―4 8. 105 x 1023 = 10―1 x 10―6 9. 100 x 10―2 = 1,000 x 106 10. 10―3 x 1023 = 10 x 1,000 Multiplying and Dividing in Exponential Notation These are the rules we use most often. 1. When multiplying and dividing using exponential notation, handle the significands and exponents separately. Do number math using number rules, and exponential math using exponential rules, then combine the two parts. Memorize rule 1, then apply it to the following three problems. a. Do not use a calculator: (2 x 103) (4 x 1023) = ***** For numbers, use number rules. 2 times 4 is 8 For exponentials, use exponential rules. 103 x 1023 = 103+23 = 1026 Then combine the two parts: (2 x 103) (4 x 1023) = 8 x 1026 b. Do the significand math on a calculator but the exponential math in your head for (2.4 x 10―3) (3.5 x 1023) = ***** Handle significands and exponents separately. • Use a calculator for the numbers. • Do the exponentials in your head. 10―3 x 1023 = 1020 • Then combine. 2.4 x 3.5 = 8.4 (2.4 x 10―3) (3.5 x 1023) = (2.4 x 3.5) x (10―3 x 1023) = 8.4 x 1020 We will review how much to round answers in Module 3. Until then, round numbers and significands in your answers to two digits unless otherwise noted. c. Do significand math on a calculator but exponential math without a calculator. 6.5 x 1023 = 4.1 x 10―8 ***** Answer: ? 6.5 x 1023 = 6.5 x 1023 = 1.585 x [1023 ― (―8) ] = 1.6 x 1031 4.1 10―8 4.1 x 10―8 ©2010 www.ChemReview.Net v. u1 Page 12 Module 1 – Scientific Notation 2. When dividing, if an exponential term does not have a significand, add a 1 x in front of the exponential so that the number-number division is clear. Try the rule on this problem. Do not use a calculator. = 10―14 ―8 2.0 x 10 ***** 10―14 = ―8 2.0 x 10 Answer: 1 2.0 x x 10―14 10―8 = 0.50 x 10―6 = 5.0 x 10―7 Practice C: Do these in your notebook. Do the odds first, then the evens if you need more practice. Try first without a calculator, then check your mental arithmetic with a calculator if needed. Write final answers in scientific notation, rounding significands to two digits. Answers are at the end of the lesson. 1. (2.0 x 101) (6.0 x 1023) = 3. 5. 3.0 x 10―21 ― 2.0 x 103 = 10―14 ― 5.0 x 10―3 2. (5.0 x 10―3) (1.5 x 1015) = 4. = 6.0 x 10―23 2.0 x 10―4 6. = 1014 4.0 x 10―4 = 7. Complete the three problems in the pretest at the beginning of this lesson. Study Summary In your problem notebook, write a list of rules in Lesson 1B that were unfamiliar, need reinforcement, or you found helpful. Then condense your list. Add this new list to your numbered points from Lesson 1A. Write and recite the combined list until you can write all of the points from memory. Then do the problems below. Practice D Start by doing every other letter. If you get those right, go to the next number. If not, do a few more of that number. Save one part of each question for your next study session. 1. Try these without a calculator. Convert final answers to scientific notation. a. 3 x (6.0 x 1023) = b. 1/2 x (6.0 x 1023) = c. 0.70 x (6.0 x 1023) = d. 103 x (6.0 x 1023) = e. 10―2 x (6.0 x 1023) = f. (― 0.5 x 10―2)(6.0 x 1023) = ©2010 www.ChemReview.Net v. u1 Page 13 Module 1 – Scientific Notation g. 1 = 12 10 h. i. 3.0 x 1024 6.0 x 1023 k. 1.0 x 10―14 = 4.0 x 10―5 1/10―9 = j. 2.0 x 1018 = 6.0 x 1023 l. = 1010 = ―5 2.0 x 10 2. Use a calculator for the numbers, but not for the exponents. a. 2.46 x 1019 = 6.0 x 1023 b. 10―14 0.0072 = Try problems 3 and 4 without using a calculator. 3. 107 x 10―2 = 10 x 10―5 4. 10―23 x 10―5 = 10―5 x 100 5. Convert to scientific notation in the final answer. Do not round during these. a. ( 74 x 105 ) + ( 4.09 x 107 ) = b. (5.122 x 10―9 ) ― ( ― 12,914 x 10―12 ) = ANSWERS 1. 1.2 x 1020 Pretest. In scientific notation: 2. 3.3 x 1028 3. 2.83 x 10―16 Practice A: You may do the arithmetic in any way you choose that results in these final answers. 1. 64.202 x 1023 = + 13.2 x 1021 64.202 x 1023 + 0.132 x 1023 64.334 x 1023 = 6.4334 x 1024 2. (61 x 10―7) + (2.25 x 10―5) + (212.0 x 10―6) = (0.61 x 10―5) + (2.25 x 10―5) + (21.20 x 10―5) = 0.61 2.25 + 21.20 24.06 3. x x x x 10―5 10―5 10―5 10―5 (10―5 is the highest value of the three exponentials) = 2.406 x 10―4 (― 54 x 10―20 ) + ( ― 2.18 x 10―18 ) = (― 0.54 x 10―18 ) + ( ― 2.18 x 10―18 ) = ─ 0.54 x 10―18 ─ 2.18 x 10―18 ─ 2.72 x 10―18 ©2010 www.ChemReview.Net v. u1 ( 10―18 is higher in value than 10―20 ) Page 14 Module 1 – Scientific Notation 4. (― 21.46 x 10―17 ) ― ( ― 3,250 x 10―19 ) = ( + 3,250 x 10―19 ) ― ( 21.46 x 10―17 ) = ( + 32.50 x 10―17 ) ― ( 21.46 x 10―17 ) = 32.50 x 10―17 21.46 x 10―17 11.04 x 10―17 ─ = 1.104 x 10―16 Practice B 1. 108 9. 2. 10―11 3. 10―1 4. 10―8 5. 104 100 x 10―2 = 102 x 10―2 = 100 = 10―9 103 x 106 109 1,000 x 106 10. 6. 10―23 7. 104 8. 1035 10―3 x 1023 = 1020 = 1016 10 x 1,000 104 (For 9 and 10, you may use different steps, but you must arrive at the same answer.) Practice C 1. 1.2 x 1025 2. 7.5 x 1012 3. ― 1.5 x 10―24 4. 3.0 x 10―19 5. ― 2.0 x 10―12 6. 2.5 x 1017 Practice D 1a. 3 x (6.0 x 1023) = 18 x 1023 = 1.8 x 1024 1b. 1/2 x (6.0 x 1023) = 3.0 x 1023 1c. 0.70 x (6.0 x 1023) = 4.2 x 1023 1d. 103 x (6.0 x 1023) = 6.0 x 1026 1e. 10―2 x (6.0 x 1023) = 6.0 x 1021 1g. 1f. (― 0.5 x 10―2)(6.0 x 1023) = ― 3.0 x 1021 10―12 1 = 12 10 1h. 1i. 3.0 x 1024 = 3.0 x 1024 6.0 x 1023 6.0 x 1023 1/10―9 = 109 = 0.50 x 101 = 5.0 x 100 ( = 5.0 ) 1j. 2.0 x 1018 = 0.33 x 10―5 = 3.3 x 10―6 6.0 x 1023 1 x 1010 2.0 x 10―5 1k. 1.0 x 10―14 = 0.25 x 10―9 = 2.5 x 10―10 4.0 x 10―5 = 0.50 x 1015 = 5.0 x 1014 1l. 1010 2.0 x 10―5 2a. 2.46 x 1019 = 0.41 x 10―4 = 4.1 x 10―5 6.0 x 1023 2b. 10―14 0.0072 3. 107 x 10―2 = 105 = 109 10―4 101 x 10―5 5a. = = 1.0 x 10―14 = 7.2 x 10―3 1.0 x 10―14 = 0.14 x 10―11 = 1.4 x 10―12 7.2 x 10―3 ( 74 x 105 ) + ( 4.09 x 107 ) = = ( 0.74 x 107 ) + ( 4.09 x 107 ) = ©2010 www.ChemReview.Net v. u1 4. 10―23 x 10―5 10―5 x 102 = 10―25 5b. (5.122 x 10―9 ) ― ( ― 12,914 x 10―12 ) = = (5.122 x 10―9 ) + ( 12.914 x 10―9 ) = Page 15 Module 1 – Scientific Notation 0.74 x 107 + 4.09 x 107 4.83 x 107 + 5.122 x 10―9 12.914 x 10―9 18.036 x 10―9 = 1.8036 x 10―8 ***** Lesson 1C: Tips for Complex Calculations Pretest: If you can solve both problems correctly, skip this lesson. Convert your final answers to scientific notation. Check your answers at the end of this lesson. (10―9)(1015) (4 x 10―4)(2 x 10―2) 1. Solve this problem without a calculator. 2. For this problem, use a calculator as needed. = (3.15 x 103)(4.0 x 10―24) = (2.6 x 10―2)(5.5 x 10―5) ***** Choosing a Calculator If you have not already done so, please read Choosing a Calculator under Notes to the Student in the preface to these lessons. Complex Calculations The prior lessons covered the fundamental rules for exponential notation. For longer calculations, the rules are the same. The challenges are keeping track of the numbers and using the calculator correctly. The steps below will help you to simplify complex calculations, minimize data-entry mistakes, and quickly check your answers. Let’s try the following calculation two ways. (7.4 x 10―2)(6.02 x 1023) = (2.6 x 103)(5.5 x 10―5) Method 1. Do numbers and exponents separately. Work the calculation above using the following steps. a. Do the numbers on the calculator. Ignoring the exponentials, use the calculator to multiply all of the significands on top. Write the result. Then multiply all the significands on the bottom and write the result. Divide, write your answer rounded to two digits, and then check below. * * * * * (See How To Use These Lessons, Point 1, on page 1). 7.4 x 6.02 2.6 x 5.5 = 44.55 = 14.3 3.1 b. Then exponents. Starting from the original problem, look only at the powers of 10. Try to solve the exponential math “in your head” without the calculator. Write the answer for the top, then the bottom, then divide. ***** ©2010 www.ChemReview.Net v. u1 Page 16 Module 1 – Scientific Notation 10―2 x 1023 = 1021 = 1021―(―2) = 1023 10―2 103 x 10―5 c. Now combine the significand and exponential and write the final answer. ***** 3.1 x 1023 Note that by grouping the numbers and exponents separately, you did not need to enter the exponents into your calculator. To multiply and divide powers of 10, you simply add and subtract whole numbers. Let’s try the calculation a second way. Method 2. All on the calculator. Enter all of the numbers and exponents into your calculator. (Your calculator manual, which is usually available online, can help.) Write your final answer in scientific notation. Round the significand to two digits. ***** On most calculators, you will need to use an E or EE or EXP key, rather than the times key, to enter a “10 to a power” term. If you needed that hint, try again, and then check below. * **** Note how your calculator displays the exponential term in answers. The exponent may be set apart at the right, sometimes with an E in front. Your calculator answer should be the same as with Method 1: 3.1 x 1023 . Which way was easier? “Numbers, then exponents,” or “all on the calculator?” How you do the arithmetic is up to you, but “numbers, then exponents” is often quicker and easier. Using the Reciprocal Key On your calculator, the reciprocal key, 1/x or x-1 , may save time and steps. Try the calculation below this way: Multiply the top. Write the answer. Multiply the bottom. Write the answer. Then divide and write your final answer. 74 x 4.09 = = 42 x 6.02 An alternative to this ”top then bottom” method is “bottom, 1/x , top.” On the calculator, repeat the above calculation using these steps. • Multiply the bottom numbers first. • Press the 1/x or x-1 button or function on your calculator. • Then multiply that result by the numbers on top. Try it. ***** ©2010 www.ChemReview.Net v. u1 Page 17 Module 1 – Scientific Notation You should get the same answer (1.197 = 1.2). On some calculators, using the 1/x or x-1 key may allow you to avoid writing or re-entering the “top-over-bottom” middle step. Your calculator manual can help with using the 1/x function. The algebra that explains why this works is 1 x A x B = (C x D)―1 x A x B AxB = CxD CxD The reciprocal key “brings the bottom of a fraction to the top.” Power of 10 Reciprocals A reciprocal method can also be used for powers of 10. For example, try the following calculation without a calculator. First do the math in your head for the top terms and write the answer. Then evaluate the denominator in your head and write the answer. Divide top by bottom in your head to get the final answer. 10―4 x 1023 = 102 x 10―7 = Now write the calculation “in your head, ” without a calculator, using these steps. • Multiply the bottom terms (by adding the bottom exponents). • “Bring the bottom exponential to the top” by changing its sign. • Multiply that result by the top terms (by adding all of the exponents). Write the final answer. ***** The steps are bottom = 2 + (― 7) = ― 5 top = + 5 ― 4 + 23 = 24 Why does “bringing an exponent up” change its sign? answer = 1024 1/10x = (10x)―1 = 10―x When you take an exponential term to a power, you multiply the exponents. (We will do more with exponents to a power in a later lesson.) For simple fractions of exponential terms, if your mental arithmetic is good, you should be able to calculate the final answer for the powers of 10 without writing down middle steps. For longer calculations, however, writing the separate “top and bottom” answers, then dividing, will break the problem into pieces that are easier to manage and check. Checking Calculator Results Whenever a complex calculation is done on a calculator, you must do the calculation a second time, using different steps, to catch errors in calculator use. Calculator results can be checked either by using a different key sequence or by estimating answers. Below is a method that uses estimation to check multiplication and division of exponential notation. Let’s use the calculation that was done in the first section of this lesson as an example. (7.4 x 10―2)(6.02 x 1023) = (2.6 x 103)(5.5 x 10―5) ©2010 www.ChemReview.Net v. u1 Page 18 Module 1 – Scientific Notation Try these steps on the above calculation. 1. Estimate the numbers first. Ignoring the exponentials, round and then multiply all of the top significands, and write the result. Round and multiply the bottom significands. Write the result. Then write a rounded estimate of the answer when you divide those two numbers, and then check below. ***** Your rounding might be 7x6 = 7 3 3x6 ≈2 (the ≈ sign means approximately equals) If your mental arithmetic is good, you can estimate the number math on the paper without a calculator. The estimate needs to be fast, but does not need to be exact. If needed, evaluate the rounded numbers on the calculator. 2. Evaluate the exponents. The exponents are simple whole numbers. Try the exponential math without the calculator. ***** 10―2 x 1023 = 1021 = 1021― (―2) = 1023 103 x 10―5 10―2 3. Combine the estimated number and exponential answers. Compare this estimated answer to answer found when you did this calculation in the section above using a calculator. Are they close? ***** The estimate is 2 x 1023. The answer with the calculator was 3.1 x 1023. Allowing for rounding, the two results are close. If your fast, rounded, “done in your head” answer is close to the calculator answer, it is likely that the calculator answer is correct. If the two answers are far apart, check your work. 4. Estimating number division. If you know your multiplication tables, and if you memorize these simple decimal equivalents to help in estimating division, you may be able to do many number estimates without a calculator. 1/2 = 0.50 1/3 = 0.33 1/4 = 0.25 1/5 = 0.20 2/3 = 0.67 3/4 = 0.75 1/8 = 0.125 The method used to get your final answer should be slow and careful. Your checking method should use different steps or calculator keys, or, if time is a factor, should use rounded numbers and quick mental arithmetic. On timed tests, you may want to do the exact calculation first, and then go back at the end, if time is available, and use rounded numbers as a check. Your skills at both estimating and finding alternate calculator steps will improve with practice. When doing a calculation the second time, try not to look back at the first answer until after you write the estimate. If you look back, by the power of suggestion, you will often arrive the first answer whether it is correct or not. ©2010 www.ChemReview.Net v. u1 Page 19 Module 1 – Scientific Notation For complex operations on a calculator, do each calculation a second time using rounded numbers and/or different steps or keys. Practice For problems 1-4, you will need to know the “changing fractions to decimal equivalent” equalities in the box above. If you need practice, try this. • On a sheet of paper, draw 5 columns and 7 rows. List the fractions down the middle column. • Write the decimal equivalents of the fractions at the far right. • Fold over those answers and repeat at the far left. Fold over those and repeat. 1/2 1/3 1/4 … Then try these next four without a calculator. Convert final answer to scientific notation. 4 x 103 = (2.00)(3.0 x 107) 1. 2. 1 = (4.0 x 109)(2.0 x 103) 3. (3 x 10―3)(8.0 x 10―5) = (6.0 x 1011)(2.0 x 10―3) 4. (3 x 10―3)(3.0 x 10―2) = (9.0 x 10―6)(2.0 x 101) For the following calculations 5-8, • First write an estimate based on rounded numbers, then exponentials. Try to do this estimate without using a calculator. • Then calculate a more precise answer. You may o plug the entire calculation into the calculator, or o use the “numbers on calculator, exponents on paper” method, or o experiment with both approaches to see which is best for you. Convert both the estimate and the final answer to scientific notation. Round the significand in the answer to two digits. Use the calculator that you will be allowed to use on quizzes and tests. To start, try every other problem. If you get those right, go to the next lesson. If you need more practice, do more. 5. (3.62 x 104)(6.3 x 10―10) = (4.2 x 10―4)(9.8 x 10―5) ©2010 www.ChemReview.Net v. u1 6. 10―2 = (750)(2.8 x 10―15) Page 20 Module 1 – Scientific Notation 7. (1.6 x 10―3)(4.49 x 10―5) = (2.1 x 103)(8.2 x 106) 8. 1 = (4.9 x 10―2)(7.2 x 10―5) 9. For additional practice, do the two pretest problems at the beginning of this lesson. ANSWERS Pretest: 1. 1.25 x 1011 2. 8.8 x 10―15 Practice: You may do the arithmetic using different steps than below, but you must get the same answer. 4 x 103― 7 = 6 1. 4 x 103 = (2.00)(3.0 x 107) 2. 1 9)(2.0 x 103) (4.0 x 10 3. ( 3 x 10―3 )(8.0 x 10―5) = ( 2 6.0 x 1011)(2.0 x 10―3) 4. = = 1 12 8 x 10 2 x 10―4 = 0.67 x 10―4 = 6.7 x 10―5 3 1 x 10―12 = 0.125 x 10―12 = 1.25 x 10―13 8 8 x 10―3―5 = 2 x 10―8 = 2 x 10―8―8 = 2.0 x 10―16 4 1011―3 108 ( 3 x 10―3 )( 3.0 x 10―2 ) = 9 x 10―3―2 = 0.50 x 10―5 ( 9.0 x 10―6 )( 2.0 x 101 ) 18 10―6+1 10―5 = 0.50 = 5.0 x 10―1 5. First the estimate. The rounding for the numbers might be 4 x 6 = 0.6 For the exponents: 104 x 10―10 = 10―6 = 109 x 10―6 = 103 10―4 x 10―5 10―9 4 x 10 ≈ 0.6 x 103 ≈ 6 x 102 (estimate) in scientific notation. For the precise answer, doing numbers and exponents separately, (3.62 x 104)(6.3 x 10―10) = 3.62 x 6.3 = 0.55 (4.2 x 10―4)(9.8 x 10―5) 4.2 x 9.8 The exponents are done as in the estimate above. = 0.55 x 103 = 5.5 x 102 (final) in scientific notation. This is close to the estimate, a check that the more precise answer is correct. 6. Estimate: 1 ≈ 1 = 0.05 ; 7 x 3 20 10―2 = 10―2― (―13) = 1011 (102)(10―15) 0.05 x 1011 = 5 x 109 (estimate) Numbers on calculator: = 0.048 1 7.5 x 2.8 FINAL: 0.048 x 1011 = 4.8 x 109 Exponents – same as in estimate. (close to the estimate) 7. You might estimate, for the numbers first, 1.6 x 4.49 = 2 x 4 = 0.5 For the exponents: 10―3 x 10―5 = 10―8 = 10―17 2.1 x 8.2 2x8 103 x 106 109 = 0.5 x 10―17 = 5 x 10―18 (estimate) ©2010 www.ChemReview.Net v. u1 Page 21 Module 1 – Scientific Notation More precisely, using numbers then exponents, with numbers on the calculator, 1.6 x 4.49 = 0.42 2.1 x 8.2 The exponents are done as in the estimate above. 0.42 x 10―17 = 4.2 x 10―18 8. Estimate: This is close to the estimate. Check! 1 ≈ 1 ≈ 0.033 ; 5x7 35 1 = 1/(10―7) = 107 (10―2)(10―5) 0.033 x 107 ≈ 3 x 105 (estimate) Numbers on calculator = 1 = 0.028 4.9 x 7.2 Exponents – see estimate. FINAL: 0.028 x 107 = 2.8 x 105 (close to the estimate) ***** Lesson 1D: The Elements – Part 1 Cognitive science has found that quick, automatic recall of core knowledge is essential when solving math and science problems. At the core of chemistry are the elements: the building 1A 2A 3A 4A 5A 6A 7A 8A 1 2 blocks of matter. There H He are 91 elements Hydrogen Helium 3 4 5 6 7 8 9 10 found in the earth’s crust. Li Be B C N O F Ne Boron Carbon Nitrogen Oxygen Fluorine Neon Lithium Beryllium The Periodic 11 12 Table helps in Na Mg predicting the Sodium Magnesium properties of the elements. In first-year chemistry, about 40 of the elements are frequently encountered in problems. Periodic Table To begin to learn those about 40 elements, your assignment is: • For the 12 elements above, memorize the name, symbol, atomic number (the integer), and the position of the element in the table. • For each element, given any one (its symbol, name, or number), be able to write the others. • Be able to fill in an empty chart like the one above with these element names, symbols, and atomic numbers. ***** ©2010 www.ChemReview.Net v. u1 Page 22 Module 2 – The Metric System SUMMARY –Scientific Notation 1. When writing a number between ─1 and 1, place a zero in front of the decimal point. Do not write .42 or ─ .74 ; do write 0.42 or ─ 0.74 2. Exponential notation represents numeric values in three parts: • a sign in front showing whether the value is positive or negative; • a number (the significand); • times a base taken to a power (the exponential term). 3. In scientific notation, the significand must be a number that is 1 or greater, but less than 10, and the exponential term must be 10 to a whole-number power. This places the decimal point in the significand after the first number which is not a zero. 4. When moving a decimal in exponential notation, the sign in front never changes. 5. To keep the same numeric value when moving the decimal of a number in base 10 exponential notation, if you • move the decimal Y times to make the significand larger, make the exponent smaller by a count of Y; • move the decimal Y times to make the significand smaller, make the exponent larger by a count of Y. Recite and repeat to remember: When moving the decimal, for the numbers after the sign in front, “If one gets smaller, the other gets larger. If one gets larger, the other gets smaller.” 6. To add or subtract exponential notation by hand, all of the values must be converted to have the same exponential term. • Convert all of the values to have the same power of 10. • List the significands and exponential in columns. • Add or subtract the significands. • Attach the common exponential term to the answer. 7. In multiplication and division using scientific or exponential notation, handle numbers and exponential terms separately. Recite and repeat to remember: • Do numbers by number rules and exponents by exponential rules. • When you multiply exponentials, you add the exponents. • When you divide exponentials, you subtract the exponents. • When you take an exponential term to a power, you multiply the exponents. • To take the reciprocal of an exponential, change the sign of the exponent. 8. In calculations using exponential notation, try the significands on the calculator but the exponents on paper. 9. For complex operations on a calculator, do each calculation a second time using rounded numbers and/or different steps or keys. #### ©2010 www.ChemReview.Net v. u1 Page 23 ...
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