SCIENTIFIC NOTATION. Scientific Notation

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1 HFCC Math Lab Beginning Algebra - SCIENTIFIC NOTATION There is a way of writing numbers that is called scientific notation. Every number that can be written as a terminating decimal can be also written in scientific notation. Numbers written in scientific notation have the following form. Scientific Notation A positive number is written in scientific notation if it is written as the product of a number a, where a is between 1 and 1, including 1 but excluding 1, and an integer power n of 1. Thus it will have the following form: a 1 n. Scientific notation is especially useful for working with very large or very small numbers. To Write a Number in Scientific Notation, Step 1. Move the decimal point in the original number to the left or right so that the new number has a value between 1 and 1, including 1 but excluding 1. Step. Count the number of decimal places the decimal point is moved in Step 1. If the original number is 1 or greater, the count is positive. If the original number is between 1 and 1, including 1 but excluding 1, then the count is zero. If the original number is less than 1, the count is negative. Step 3. Multiply the new number in Step 1 by 1 raised to an exponent equal to the count found in Step. Ex 1: Write 8, in scientific notation. Step 1 Move the decimal point until the number is between 1 and 1. 8 ss, sss. places Thus a= 8. Step As the decimal point was moved places and the original number is 1 or greater, the count is positive. Hence n=. Step 3 8, = 8. 1 is the answer in scientific notation. Revised /9 1

2 Ex : Write. in scientific notation. Step 1 Move the decimal point until the number is between 1 and 1..rr places Thus a=. Step As the decimal point was moved places and the original number is less than 1, the count is negative. Hence n= Step 3.=. 1 is the answer in scientific notation. Ex 3: Write 93 in scientific notation. Step 1 Move the decimal point until the number is between 1 and 1. 9 ss 3. places Thus a= 9.3 Step As the decimal point was moved places and the original number is 1 or greater, the count is positive. Hence n= Step 3 93= is the answer in scientific notation. Ex 4: Write.1 in scientific notation. Step 1 Move the decimal point until the number is between 1 and 1..rrrr 1r places Thus a= 1. Step As the decimal point was moved places, and the original number is less than 1, the count is negative. Hence n= Step 3.1= 1. 1 is the answer in scientific notation. Ex : Write 7.98 in scientific notation. Step 1 This step is done as the number is already between 1 and 1. Step The count is zero as you did not need to move the decimal point. Step = is the answer in scientific notation. Revised /9

3 However, what do you do if you wish to convert a number in scientific notation to a number in standard form? How do we go about doing this? To Convert a Number from Scientific Notation to Standard Form, Move the decimal point the same number of places as the exponent of 1. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left. If the exponent is zero, do not move the decimal point. Ex : Write. 1 in standard form. Note that the exponent of 1 is. As it is negative, you must move the decimal point places to the left. sssss. s left places. 1. = is the answer in standard form. Ex 7: Write in standard form. Note that the exponent of 1 is 4. As it is positive, you must move the decimal point 4 places to the right. 3.7rrrr right 4 places = 37, is the answer in standard form. Ex 8: Write.3 1 in standard form. Note that the exponent of 1 is. As it is zero, you do not move the decimal point..3 1 =.3 is the answer in standard form. Revised /9 3

4 Ex 9: Write in standard form. Note that the exponent of 1 is 4. As it is negative, you must move the decimal point 4 places to the left. ssss 7.89 left 4 places = is the answer in standard form. Ex 1: Write in standard form. Note that the exponent of 1 is. As it is positive, you must move the decimal point places to the right. 4.rrr 81r rr right places = 4, 8,1 is the answer in standard form. How do you multiply or divide numbers written in scientific notation? m n m n First recall that the product rule for exponents is x x = x + m x m n while the quotient rule for exponents is = x n x To Multipy or Divide Numbers in Scientific Notation, Multiply or divide the numbers, a, that are between 1 and 1, including 1. Then multiply or divide the powers of 1 using your product rule or quotient rule for exponents. Make sure that your final number is in scientific notation. If it is not in scientific notation, convert it to scientific notation. Ex 11: Simplify ( )( ) using the associative and commutative laws Revised /9 4

5 9 3 Ex 1: Simplify ( )(.4 1 ) using the associative and commutative laws adding exponents as the bases are the same note 3.38 is not between 1 and converting 3.38 to scientific notation Ex 13: Simplify using the associative and commutative laws.4 1 subtracting exponents as the bases are the same Ex 14: Simplify using the associative and commutative laws subtracting exponents as the bases are the same note.937 is not between 1 and converting.937 to scientific notation adding exponents as the bases are the same Revised /9

6 EXERCISES Write each of the following numbers in scientific notation: , , , ,133,1 1. 4, ,, Write each of the following numbers in standard form: Multiply or divide, and write scientific notation for the result: ( )( ) 3. ( )( 8. 1 ) ( )( 4. 1 ) Revised /9

7 rrr 1r 4 7 thus a= places SOLUTIONS AND ANSWERS TO EXERCISES Count = 4 as the original number is less than 1 Hence n= 4.147= ,9,9s s. s thus a=.9 3 places Count = 3 as the original number is 1 or greater. Hence n= 3. 3, 9= rr 3 thus a= 3. places Count = as the original number is less than 1. Hence n=..3= ,3 7 ss,3. sss thus a= 7.3 places Count = as the original number is 1 or greater. Hence n=. 7, 3= r thus a=. 1 place Count = 1 as the original number is less than 1. Hence n= 1..=. 1 1 Revised /9 7

8 s thus a= place Count = 1 as the original number is 1 or greater. Hence n= = or you can write it as as you don t write an exponent of rrr 7 9 thus a= places Count = 3 as the original number is less than 1. Hence n= 3..79= ,31 4, s ss 31. s thus a= places Count = 4 as the original number is greater than 1. Hence n= , 31= ,, 1ss, sss, sss. thus a= 1 8 places Count = 8 as the original number is greater than 1. Hence n= ,, = Note that the exponent of 1 is. As it is positive, you must move the decimal point places to the right. 1.7r r right places = 17 Revised /9 8

9 Note that the exponent of 1 is. As it is negative, you must move the decimal point places to the left. 9.8 ss left places =.98., Note that the exponent of 1 is. As it is positive, you must move the decimal point places to the right..rrrrrr right places. 1 =,, 4. 83,3, Note that the exponent of 1 is. As it is negative, you must move the decimal point places to the left. ssssss 9.8 left places = Note that the exponent of 1 is 1. As it is positive, you must move the decimal point 1 place to the right..17 r right 1 place = 1.7 Revised /9 9

10 Note that the exponent of 1 is 1. As it is positive, you must move the decimal point 1 places to the left.. sssssssss s left 1 places 1 1 =. 3. 3,,, 31. ( )(.3 1 ) using the associative and commutative laws ( )( 4. 1 ) using the associative and commutative laws note 13.9 is not between 1 and converting 13.9 to scientific notation using the associative and commutative laws subtracting exponents as the bases are the same note.8 is not between 1 and converting.8 to scientific notation Revised /9 1

11 NOTE: You can get additional instruction and practice by going to the following web sites: Then click on Tutorial #9: Negative Exponents and Scientific Notation for more examples on writing numbers in scientific notation and/or standard form. Scroll down to the middle of the page that opens and then look at the examples that deal with writing numbers in scientific notation or in standard form Only go through this page (page 3 out of ) of this tutorial for more examples on writing numbers in scientific notation or in standard form. Revised /9 11