Adding/Subtracting/Multiplying /Dividing Numbers in Scientific Notation
How wide is our universe?
210,000,000,000,000,000,000,0 00 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 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. 1. 2. 3. 4.
2.87509 x 10-5 2.87509 x 10-4 2.87509 x 104 2.87509 x 105
-4 10
2) Express 1.8 x 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
96780
2) 7.4521 x 10-3
.0074521
3) 8.513904567 x 107
85139045.67
4) 4.09748 x 10-5
.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 when the Exponents are Different
• 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. 1. 2. 3. 4.
14.28 x 10-4 1.428 x 10-3 14.28 x 1010 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 2 1.2 x 10 :
The answer in scientific notation is 6 x 10 -11 The answer in decimal notation is 0.00000000006
4) Evaluate: 4.5 x 10-5 1.6 x 10-2 0.0028125 Write in scientific notation. 2.8125 x 10-3
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
