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1 Success Center Directed Learning Activity (DLA) Scientific Notation M102.1 2 DLA: SCIENTIFIC NOTATION Description: After reviewing the rules for multiplying and dividing exponents, you will apply these rules to the concept of scientific notation, which, in turn, will provide you with fundamental skills necessary to succeed in science courses, such as chemistry and microbiology. Prior Knowledge: You must know the following rules of exponents. 1. Product Rule: xa xb xab . Ex. x 2 x5 x 25 x7 and 103 109 103( 9) 106. x8 xa 10 a b x . 2. Quotient Rule: b Ex. 5 x85 x3 and 101( 6) 1016 107. 6 x x 10 3. Reciprocal Rule: x a 1 1 1 1 . Ex. x 2 2 , 108 , and 105. a 8 5 x x 10 10 Materials: You will need a scientific calculator (although some instructors may require that you do this work without a calculator. Check with your instructor.) Part I: Practice Exercises Simplify the following expressions. Express your final answers without negative exponents. 1. x x 2. 3m 5m 4. 6t 5 7t 8 5. 2 5 6 27v 4 3v 9 12 y 9 3. 8 y4 6. 7m6 8m4 3 Now, check your answers. If you have any questions, see a tutor before moving on to the next section. 1. x 7 ; 2. 15m7 ; 3. 3 y5 42 9 56 ; 4. 3 ; 5. 5 ; 6. 10 2 t v m Part II. Powers of 10 Simplify the following expressions. Express your final answers as powers of 10 (the exponents may be positive, negative or zero). See examples 1 and 2. Example 1: 105 108 1058 103 Example 2: 102 1026 108 106 1. 104 108 2. 105 104 3. 10 102 105 10 5. 109 109 6. 1 106 4. Now, check your answers. If you have any questions, see a tutor before moving on to the next section. 1. 1012 ; 2. 10; 3. 103 ; 4. 106 ; 5. 100 ; 6. 106 4 Part III. Products with powers of 10 Multiplying by powers of ten is easy. You just need to know where to move the decimal point. Remember, if the exponent of 10 is positive, then you are multiplying by a number that is greater than 1, and your answer will get larger, so you must move the decimal point to the right. If the exponent of 10 is negative, then you are multiplying by a number that is less than 1, and your answer will get smaller, so you must move the decimal point to the left. Study the following examples, and then do the following multiplications. Example 1: 2.53 105 253,000 Note that the decimal point in the answer is not written, but it is assumed to be after the last zero, 5 places to the right of where it was originally. Example 2: 2.53 105 0.0000253 Note that the decimal point in the answer is five places to the left of where it was originally. 1. 3.7 104 2. 0.005 105 3. 7.1106 4. 217.4 104 Part IV. Scientific Notation A. Definition of Scientific Notation Sometimes scientists need to write very big or very small numbers that contain many zeros, such as the number of molecules of a substance in one mole of the substance, known as Avogadro Constant, which is 602,214,179,000,000,000,000,000; or, the radius of an electron, which is approximately 0.00000000000000281794 meters. Imagine having to write numbers such as these several times in a lab report!! Well, scientists use a more efficient notation known as scientific notation. A number is in scientific notation when it is in the form A10 x where A is a number greater than or equal to one and less than 10, and x is an integer (which means that the exponent cannot be a fraction or a decimal). For example, the following numbers are in scientific notation: a) 2.8 109 , b) 7.032 107 , c) 1103 , d) 9.9976 10 5 Explain why each of the following numbers is NOT in scientific notation. See examples 1 and 2. Example 1. 12.5 104 : The number 12.5 is not less than 10 Example 2. 3.14 105.4 : The exponent of 10 is not an integer 1 1. 110 2 : 2. 36 108 : 3. 0.98 106 : 1 2 4. 110 : 5. 13.25 100.7 : 6. 2.5 610 : 7. 6.408 : 8. 4.58 104 : 6 B. Standard Notation and Scientific Notation. One of the skills you need to develop is the ability to change numbers back and forth between scientific notation and standard notation. Let us practice. Write the following numbers in standard notation. See examples 3 and 4. Example 3: 2.8 105 280,000 Example 4: 5.01108 0.0000000501 9. 1.1107 10. 7 103 11. 9.03 106 12. 8.403 107 13. 7.2 104 14. 4.3786 105 Sometimes you will be required to write numbers in scientific notation. When these numbers are in standard form, there a simple 3-step process to do this: 1. Write the first digit the given number that is NOT zero and place a decimal point after this digit. 2. Copy all the remaining digits of the given number until you get to the last digit that is NOT zero. 3. Multiply by the appropriate power of ten which would move the decimal point to its original position. For example to write the number 52,300,000 in scientific notation we carry out the 3step process as follows: Step 1: 5. [The first digit that is not zero is 5] Step 2: 5.23 [The last digit that is not zero is 3] Step 3: 5.23 107 [The original position of the decimal point is 7 places to the right of where it is now, so we multiply by 107 ] So, 52,300,000 = 5.23 107 7 Now, you carry out the 3-step process to write the number 0.000008069 in scientific notation. Fill in the blanks. Step 1: _____ [The first digit that is not zero is ___ ] Step 2: _____ [The last digit that is not zero is ___ ] Step 3: ___________ [The original position of the decimal point is ___ places to the _______ of where it is now, so we multiply by _____ ] So, 0.000008069 = ________________ Now, write the following numbers in scientific notation. 15. 12,000,000,000 = 16. 0.0000005068 = 17. 0.0000000000001 = 18. 908,000,000 = 19. 0.962 = 20. 300 = 21. Avogadro Constant: 602,214,179,000,000,000,000,000 = 22. Radius of an electron: 0.00000000000000281794 meters = 8 C. Numbers in “Almost” Scientific Notation Sometimes, numbers that are not in scientific notation look like they are written in scientific notation. For example, the number 12.4 105 is not in scientific notation because 12.4 in not less than 10. The question is, how do we write such a number in scientific notation? One approach is to write the number 12.4 105 in standard form and then change it into scientific notation: 12.4 105 1, 240,000 1.24 106. Unfortunately, if the exponent of 10 is large, this method can be tedious. Imagine having to write the number 12.4 1025 in scientific notation. Using the method we used above would give us: 12.4 1025 124,000,000,000,000,000,000,000,000 1.24 1026. Can you think of a different way to get the answer without having to write the number in standard form? Stop reading at this point and think about this question now. Write your thoughts on the lines below. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ One alternative is to write the number being multiplied by a power of ten in scientific notation, and then multiply the powers of ten using the rules for exponents we reviewed at the beginning of this packet. Let’s revisit the problem of writing 12.4 1025 in scientific notation. First we write 12.4 in scientific notation: 12.4 = 1.24 101 So we have 12.4 1025 1.24 101 1025 1.24 1026. Now apply the same technique to write the number 347.4 1055 in scientific notation. Fill in the blanks. First we write 347.4 in scientific notation: 347.4 = _____ 10___ So we have 347.4 1055 _______10 ___ 1055 ______10 ___. 9 Your turn! Write the following numbers in scientific notation. 23. 37.9 1012 24. 0.45 108 25. 12.6 1030 26. 0.0045 1020 27. 195 1012 28. 195 1012 154 106 30. 29. 154 106 Part V. Multiplying and Dividing in Scientific Notation Some problems in science require you to multiply or divide numbers in scientific notation. Typically, in these problems you need to write your final answer in scientific notation. Study the following examples. Example 1. 5.2 10 4 10 5.2 4 10 6 9 6 109 20.8 1015 2.08 101 1015 2.08 1016. Example 2. 6.2 106 6.2 106 10 0.775 104 7.75 101 104 7.75 105. 10 8 10 8 10 If you are unclear on any step of the previous two examples, ask a tutor for help before you continue. 10 Do the following operations and write your answers in scientific notation. Use your calculator as needed. 1. 3.4 10 6.110 2. 7.5 10 4.9 10 3. 3.78 1021 8.4 1019 4. 9 10 9.9 10 6. 3.996 10 2.7 109 3 5. 1.9 1012 8.1 5 8 120 12 180 11 Part VI: Reflections a) Why did you (or your instructor) decide that completing this activity was a valuable learning experience? b) Prior to completing this activity, did you struggle with content in any of your classes because you did not understand scientific notation? Please be specific. Name the class and the topics with which you struggled. c) What was the most challenging part of this activity? How did you deal with this challenge? d) Name something new you learned as a result of completing this activity that you think will help you do better in any of your classes. Be specific. 12 13 M102.1 – Scientific Notation __________________________________________________ PRINT STUDENT NAME _______________________ STUDENT # Tutor Feedback: _________ Student completed the entire activity _________ Student demonstrated understanding of the process during the discussion of his/her work _________ Student was thoughtful in his reflections and expressed his/her thoughts using complete sentences and proper grammar. Additional Comments: _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ ___________________________________________________ PRINT INSTRUCTOR/TUTOR NAME ___________________ DATE INSTRUCTOR/TUTOR SIGNATURE STUDENT – DO NOT FORGET TO TURN THIS SHEET IN AT THE FRONT DESK! You may not get credit for completing this DLA if you fail to leave this sheet with the front desk receptionist.