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**Adding/Subtracting/Multiplying/Dividing Numbers in Scientific Notation**

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**How wide is our universe?**

210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

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**A number is expressed in scientific notation when it is in the form**

a x 10n where a is between 1 and 10 and n is an integer

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**An easy way to remember this is:**

If an exponent is positive, the number gets larger, so move the decimal to the right. If an exponent is negative, the number gets smaller, so move the decimal to the left.

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**When changing from Standard Notation to Scientific Notation:**

4) See if the original number is greater than or less than one. If the number is greater than one, the exponent will be positive. = x 105 If the number is less than one, the exponent will be negative. = 6.72 x 10-8

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**Write the width of the universe in scientific notation.**

210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

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2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 1023

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**Write 28750.9 in scientific notation.**

x 10-5 x 10-4 x 104 x 105

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**2) Express 1.8 x 10-4 in decimal notation.**

3) Express 4.58 x 106 in decimal notation. 4,580,000

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**Try changing these numbers from Scientific Notation to Standard Notation:**

9.678 x 104 x 10-3 x 107 x 10-5 96780

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**Write in PROPER scientific notation**

Write in PROPER scientific notation. (Notice the number is not between 1 and 10) 8) x 109 2.346 x 1011 9) x 104 6.42 x 10 2

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**Adding/Subtracting when Exponents are Equal**

When the exponents are the same for all the numbers you are working with, add/subtract the base numbers then simply put the given exponent on the 10.

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**General Formulas (N X 10x) + (M X 10x) = (N + M) X 10x**

(N X 10y) - (M X 10y) = (N-M) X 10y

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**Example 1 Given: 2.56 X 103 + 6.964 X 103 Add: 2.56 + 6.964 = 9.524**

Answer: X 103

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Example 2 Given: 9.49 X 105 – X 105 Subtract: 9.49 – = 4.627 Answer: X 105

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**Adding With the Same Exponent**

(3.45 x 103) + (6.11 x 103) = 9.56 9.56 x 103

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**Subtracting With the Same Exponent**

(8.96 x 107) – (3.41 x 107) 8.96 – 3.41 = 5.55 5.55 x 107

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**Adding/Subtracting when the Exponents are Different**

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**When adding or subtracting numbers in scientific notation, the exponents must be the same.**

If they are different, you must move the decimal either right or left so that they will have the same exponent.

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Moving the Decimal For each move of the decimal to the right you have to add -1 to the exponent. For each move of the decimal to the left you have to add +1 to the exponent.

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Continued… It does not matter which number you decide to move the decimal on, but remember that in the end both numbers have to have the same exponent on the 10.

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Example 1 Given: 2.46 X X 103 Shift decimal 3 places to the left for 103. Move: X 103+3 Add: 2.46 X X 106 Answer: X 106

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Example 2 Given: X 103 – 2.65 X 10-1 Shift decimal 4 places to the right for 10-1. Move: X 10(-1+4) Subtract: X X 103 Answer: X 103

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(4.12 x 106) + (3.94 x 104) (412 x 104) + (3.94 x 104) = x 104 Express in proper form: 4.15 x 106

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**Subtracting With Different Exponents**

(4.23 x 103) – (9.56 x 102) (42.3 x 102) – (9.56 x 102) 42.3 – 9.56 = 32.74 32.74 x 102 Express in proper form: 3.27 x 103

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**Multiplying… The general format for multiplying is as follows…**

(N x 10x)(M x 10y) = (N)(M) x 10x+y First multiply the N and M numbers together and express an answer. Secondly multiply the exponential parts together by adding the exponents together.

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Multiplying… Finally multiply the two results for the final answer. (2.41 x 104)(3.09 x 102) 2.41 x 3.09 = 7.45 4 + 2 = 6 7.45 x 106

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**7) evaluate (3,600,000,000)(23). The answer in scientific notation is**

8.28 x 10 10 The answer in decimal notation is 82,800,000,000

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**6) evaluate (0.0042)(330,000). The answer in decimal notation is 1386**

The answer in scientific notation is 1.386 x 103

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**Write (2.8 x 103)(5.1 x 10-7) in scientific notation.**

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Now it’s your turn. Use the link below to practice multiplying numbers in scientific notation. Multiplying in Scientific Notation

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**Dividing… The general format for dividing is as follows…**

(N x 10x)/(M x 10y) = (N/M) x 10x-y First divide the N number by the M number and express as an answer. Secondly divide the exponential parts by subtracting the exponent from the exponent in the upper number.

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Dividing… Finally divide the two results together to get the final answer. (4.89 x 107)/(2.74 x 104) 4.89 / 2.74 = 1.78 7 – 4 = 3 1.78 x 103

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5) evaluate: x x 102 : The answer in scientific notation is 6 x The answer in decimal notation is

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**0.0028125 Write in scientific notation. 2.8125 x 10-3**

4) Evaluate: x x 10-2 Write in scientific notation. x 10-3

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Now it’s your turn. Use the link below to practice dividing numbers in scientific notation. Dividing in Scientific Notation

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**Practice Adding and Subtracting in Scientific Notation**

Practice Worksheet Practice Adding and Subtracting in Scientific Notation Answers to Worksheet

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**Links for more information and practice**

Addition and Subtraction with Scientific Notation Problem Solving--Scientific Notation Scientific Notation

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Quiz Time!!! Below is a set of links for a quiz on adding and subtracting numbers in scientific notation, and there is a link to get the answers to the quiz. Adding and Subtracting Quiz Answers to Quiz

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