Adding/Subtracting/Multiplying/Dividing Numbers in Scientific Notation Notes to help you study

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.

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

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.

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. 348943 = 3.489 x 105 If the number is less than one, the exponent will be negative. .0000000672 = 6.72 x 10-8

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

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

Write 28750.9 in scientific notation. 2.87509 x 10-5 2.87509 x 10-4 2.87509 x 104 2.87509 x 105

2) Express 1.8 x 10-4 in decimal notation. 0.00018 3) Express 4.58 x 106 in decimal notation. 4,580,000

Try changing these numbers from Scientific Notation to Standard Notation: 9.678 x 104 7.4521 x 10-3 8.513904567 x 107 4.09748 x 10-5 96780 .0074521 85139045.67 .0000409748

Write in PROPER scientific notation Write in PROPER scientific notation. (Notice the number is not between 1 and 10) 8) 234.6 x 109 2.346 x 1011 9) 0.0642 x 104 6.42 x 10 2

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.

General Formulas (N X 10x) + (M X 10x) = (N + M) X 10x (N X 10y) - (M X 10y) = (N-M) X 10y

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

Example 2 Given: 9.49 X 105 – 4.863 X 105 Subtract: 9.49 – 4.863 = 4.627 Answer: 4.627 X 105

Adding With the Same Exponent (3.45 x 103) + (6.11 x 103) 3.45 + 6.11 = 9.56 9.56 x 103

Subtracting With the Same Exponent (8.96 x 107) – (3.41 x 107) 8.96 – 3.41 = 5.55 5.55 x 107

Adding/Subtracting when the Exponents are Different This requires more work…..

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.

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.

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.

Example 1 Given: 2.46 X 106 + 3.476 X 103 Shift decimal 3 places to the left for 103. Move: .003476 X 103+3 Add: 2.46 X 106 + .003476 X 106 Answer: 2.463 X 106

Example 2 Given: 5.762 X 103 – 2.65 X 10-1 Shift decimal 4 places to the right for 10-1. Move: .000265 X 10(-1+4) Subtract: 5.762 X 103-.000265 X 103 Answer: 5.762 X 103

(4.12 x 106) + (3.94 x 104) (412 x 104) + (3.94 x 104) 412 + 3.94 = 415.94 415.94 x 104 Express in proper form: 4.15 x 106

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

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.

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

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

6) evaluate (0.0042)(330,000). The answer in decimal notation is 1386 The answer in scientific notation is 1.386 x 103

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

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

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.

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

5) evaluate: 7.2 x 10-9 1.2 x 102 : The answer in scientific notation is 6 x 10 -11 The answer in decimal notation is 0.00000000006

0.0028125 Write in scientific notation. 2.8125 x 10-3 4) Evaluate: 4.5 x 10-5 1.6 x 10-2 0.0028125 Write in scientific notation. 2.8125 x 10-3