Adding/Subtracting/Adding/Subtracting/Multiplying/Dividing Numbers Multiplying/Dividing Numbers

in Scientific Notationin Scientific Notation

Adding/Subtracting/Adding/Subtracting/Multiplying/Dividing Numbers Multiplying/Dividing Numbers

in Scientific Notationin Scientific Notation

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.

Scientific NotationScientific NotationScientific NotationScientific Notation

A number is expressed in A number is expressed in scientific notation when it is scientific notation when it is

in the formin the form

a x 10a x 10nn

where where aa is between 1 and 10 is between 1 and 10

and and nn is an integer 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

2.10,000,000,000,000,00

0,000,000.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 is2.1 x 1023

Write 28750.9 in scientific notation.

1. 2.87509 x 10-5

2. 2.87509 x 10-4

3. 2.87509 x 104

4. 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:

1) 9.678 x 104

2) 7.4521 x 10-3

3) 8.513904567 x 107

4) 4.09748 x 10-5

96780

.0074521

85139045.67

.0000409748

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 Adding/Subtracting when the Exponents are when the Exponents are

DifferentDifferent

Adding/Subtracting Adding/Subtracting when the Exponents are when the Exponents are

DifferentDifferent

• 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

1386The answer in scientific

notation is 1.386 x 103

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

1. 14.28 x 10-4

2. 1.428 x 10-3

3. 14.28 x 1010

4. 1.428 x 1011

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

4) Evaluate: 4) Evaluate: 4.5 x 104.5 x 10-5-5

1.6 x 101.6 x 10-2-2

4) Evaluate: 4) Evaluate: 4.5 x 104.5 x 10-5-5

1.6 x 101.6 x 10-2-2

0.00281250.0028125Write in scientific notation.Write in scientific notation.

2.8125 x 102.8125 x 10-3-3

Now it’s your turn.• Use the link below to practice

dividing numbers in scientific notation.

• Dividing in Scientific Notation

Practice Worksheet

• Practice Adding and Subtracting in Scientific Notation

• Answers to Worksheet

Links for more information and

practice

• Addition and Subtraction with Scientific Notation

• Problem Solving--Scientific Notation

• Scientific Notation

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