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**PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION**

ADDITION AND SUBTRACTION

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**Review: M x 10n Scientific notation expresses a number in the form:**

n is an integer Any number between 1 and 10

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IF the exponents are the same, we simply add or subtract the numbers in front and bring the exponent down unchanged. 4 x 106 + 3 x 106 _______________ 7 x 106

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**If the exponents are NOT the same, we must move a decimal to make them the same.**

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**Determine which of the numbers has the smaller exponent.**

Change this number by moving the decimal place to the left and raising the exponent, until the exponents of both numbers agree. Note that this will take the lesser number out of standard form. Add or subtract the coefficients as needed to get the new coefficient. The exponent will be the exponent that both numbers share. Put the number in standard form.

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4.00 x 106 4.00 x 106 x 105 + .30 x 106 Move the decimal on the smaller number to the left and raise the exponent ! Note: This will take the lesser number out of standard form.

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4.00 x 106 4.00 x 106 x 105 + .30 x 106 4.30 x 106 Add or subtract the coefficients as needed to get the new coefficient. The exponent will be the exponent that both numbers share.

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**Make sure your final answer is**

in scientific notation. If it is not, convert is to scientific notation.!

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A Problem for you… 2.37 x 10-6 x 10-4

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Solution… x 10-6 2.37 x 10-6 x 10-4

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Solution… x 10-4 x 10-4 x 10-4

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**PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION**

MULTIPLYING AND DIVIDING

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**Rule for Multiplication**

When multiplying with scientific notation: Multiply the coefficients together. Add the exponents. The base will remain 10.

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(2 x 103) • (3 x 105) = 6 x 108

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(4.6x108) (5.8x106) =26.68x1014 Notice: What is wrong with this example? Although the answer is correct, the number is not in scientific notation. To finish the problem, move the decimal one space left and increase the exponent by one. 26.68x1014 = 2.668x1015

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((9.2 x 105) x (2.3 x 107) = 21.16 x 1012 = 2.116 x 1013

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(3.2 x 10-5) x (1.5 x 10-3) = 4.8 • 10-8

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Rule for Division When dividing with scientific notation Divide the coefficients Subtract the exponents. The base will remain 10.

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(8 • 106) ÷ (2 • 103) = 4 x 103

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**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) =

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**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

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**Please divide the following numbers.**

(5.76 x 102) / (4.55 x 10-4) = (3 x 105) / (7 x 104) = (5.63 x 108) / (2) = (8.2 x 10-6) / (9.4 x 10-3) = (4.55 x 10-14) / (3.77 x 1011) =

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**Please divide the following numbers.**

(5.76 x 102) / (4.55 x 10-4) = 1.27 x 106 (3 x 105) / (7 x 104) = 4.3 x 100 = 4.3 (5.63 x 108) / (2 x 100) = x 108 (8.2 x 10-6) / (9.4 x 10-3) = 8.7 x 10-4 (4.55 x 10-14) / (3.77 x 1011) = 1.2 x 10-25

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**PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION**

Raising Numbers in Scientific Notation To A Power

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(5 X 104)2 = (5 X 104) X (5 X 104) = (5 X 5) X (104 X 104) = (25) X 108 = 2.5 X 109

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**Try These: (3.45 X 1010)2 (4 X 10-5)2 (9.81 X 1021)2 1.19 X 1021**

(3.45 X 1010)2 = (3.45 X 3.45) X (1010 X 1010) = (11.9) X (1020) = 1.19 X 1021 (4 X 10-5)2 = (4 X 4) X (10-5 X 10-5) = (16) X (10-10) = 1.6 X 10-9 (9.81 X 1021)2 = (9.81 X 9.81) X (1021 X 1021) = (96.24) X (1042) = X 1043

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**Scientific Notation Makes These Numbers Easy**

9.54x107 miles 1.86x107 miles per second

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