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PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION ADDITION AND SUBTRACTION Review: Scientific notation expresses a number in the form: M x 1 M 10 n 10 n is an integer 6 10 4 x 6 + _______________ 3 x 10 7 x 106 IF the exponents are the same, we simply add or subtract the numbers in front and bring the exponent down unchanged. 6 10 4 x 6 + _______________ 3 x 10 7 x 106 IF the exponents are the same, we simply add or subtract the numbers in front and bring the exponent down unchanged. 106 4 x + 3 x 105 If the exponents are NOT the same, we must move a decimal to make them the same. 6 10 4.00 x 4.00 x 6 5 + .30 x 10 + 3.00 x 10 6 4.30 x 10 Move the decimal on the smaller number! 6 10 A Problem for you… -6 10 2.37 x -4 + 3.48 x 10 Solution… -6 002.37 2.37 x 10 -4 + 3.48 x 10 Solution… -4 0.0237 x 10 -4 + 3.48 x 10 -4 3.5037 x 10 2.0 x 102 + 3.0 x 103 .2 x 103 + 3.0 x 103 = .2+3 x 103 = 3.2 x 103 Addition and subtraction Scientific Notation 1. Make exponents of 10 the same 2. Add 0.2 + 3 and keep the 103 intact The key to adding or subtracting numbers in Scientific Notation is to make sure the exponents are the same. 2.0 x 107 - 6.3 x 105 2.0 x 107 -.063 x 107 = 2.0-.063 x 107 = 1.937 x 107 1. Make exponents of 10 the same 2. Subtract 2.0 - .063 and keep the 107 intact PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION MULTIPLYING AND DIVIDING Rule for Multiplication When multiplying with scientific notation: 1.Multiply the coefficients together. 2.Add the exponents. 3.The base will remain 10. (2 x 103) • (3 x 105) = 6 x 108 (9.2 x 105) x (2.3 x 107) = 21.16 x 1012 = 2.116 x 1013 (3.2 x 10-5) x (1.5 x 10-3) = 4.8 • 10-8 Rule for Division When dividing with scientific notation 1.Divide the coefficients 2.Subtract the exponents. 3.The base will remain 10. 6.20 x 10 – 5 8.0 x 10 3 6.20 8.0 = 0.775 x 10 10 -5 10 3 DIVIDE USING SCIENTIFIC NOTATION 1. Divide the # ’s & Divide the powers of ten (subtract the exponents) 2. Put Answer in Scientific Notation -8 = 7.75 x 10 – 9 (8 • 106) ÷ (2 • 103) = 4 x 103 (3.402 x 105) ÷ (6.3 x 107) = 0.54 x 10-2 Please multiply the following numbers. (5.76 x 102) x (4.55 x 10-4) = (3 x 105) x (7 x 104) = (5.63 x 108) x (2 x 100) = (4.55 x 10-14) x (3.77 x 1011) = (8.2 x10-6) x (9.4 x 10-3) = Please multiply the following numbers. (5.76 x 102) x (4.55 x 10-4) = 2.62 x 10-1 (3 x 105) x (7 x 104) = 2.1 x 1010 (5.63 x 108) x (2 x 100) = 1.13 x 109 (4.55 x 10-14) x (3.77 x 1011) = 1.72 x 10-2 (8.2 x10-6) x (9.4 x 10-3) = 7.71 x 10-8 Please divide the following numbers. 1. (5.76 x 102) / (4.55 x 10-4) = 2. (3 x 105) / (7 x 104) = 3. (5.63 x 108) / (2) = 4. (8.2 x 10-6) / (9.4 x 10-3) = 5. (4.55 x 10-14) / (3.77 x 1011) = Please divide the following numbers. 1. (5.76 x 102) / (4.55 x 10-4) = 1.27 x 106 2. (3 x 105) / (7 x 104) = 4.3 x 100 = 4.3 3. (5.63 x 108) / (2 x 100) = 2.82 x 108 4. (8.2 x 10-6) / (9.4 x 10-3) = 8.7 x 10-4 5. (4.55 x 10-14) / (3.77 x 1011) = 1.2 x 10-25 Changing from Standard Notation to Scientific Notation Ex. 6800 1. Move decimal to get a single digit # and count places moved 6800 3 2 1 3 68 x 10 Ex. 4.5 x 10 00045 2. Answer is a single digit number times the power of ten of places moved. 3 2 1 -3 Changing from Scientific Notation to Standard Notation 1. Move decimal the same number of places as the exponent of 10. (Right if Pos. Left if Neg .) 9.54x107 miles If the decimal is moved left the power is positive. If the decimal is moved right the power is negative. 1.86x107 miles per second What is Scientific Notation Multiply two numbers in Scientific Notation (3 x 10 4 )(7 x 10 –5 ) 4 = (3 x 7)(10 x 10 –5 ) 1. = 21 x 10 -1 = 2.1 x 10 2. 3. 4. 0 or 2.1 6.20 x 10 – 5 8.0 x 10 3 6.20 10 -5 8.0 10 3 = 0.775 x 10 Put # ’s in ( ) ’s Put base 10 ’s in ( ) ’s Multiply numbers Add exponents of 10. Move decimal to put Answer in Scientific Notation 3 or 2.0 x 10 2 Why do we use it? It’s a shorthand way of writing very large or very small numbers used in science and math and anywhere we have to work with very large or very small numbers. DIVIDE USING SCIENTIFIC NOTATION .2 x 10 3 + 3.0 x 10 + 3.0 x 10 = 3.2 x 10 1. 2. Divide the # ’s & Divide the powers of ten (subtract the exponents) Put Answer in Scientific Notation 7 2.0 x 10 2.0 x 10 7 3 Addition and subtraction Scientific Notation 1. Make exponents of 10 the same 2. Add 0.2 + 3 and keep the 10 3 intact The key to adding or subtracting numbers in Scientific Notation is to make sure the exponents are the same. 3 - 6.3 x 10 -.063 x 10 = 2.0-.063 x 10 = 1.937 x 10 3 3 = .2+3 x 10 -8 = 7.75 x 10 – 9 A number expressed in scientific notation is expressed as a decimal number between 1 and 10 multiplied by a power of 10 ( eg , 7000 = 7 x 10 -6 0.0000019 = 1.9 x 10 ) Scientific Notation Makes These Numbers Easy 7 7 5 7 1. Make exponents of 10 the same 2. Subtract 2.0 - .063 and keep the 10 7 intact