Download Scientific Notation

1
Success Center
Directed Learning Activity (DLA)
Scientific
Notation
M102.1
2
DLA: SCIENTIFIC NOTATION
Description: After reviewing the rules for multiplying and dividing exponents, you will apply
these rules to the concept of scientific notation, which, in turn, will provide you with
fundamental skills necessary to succeed in science courses, such as chemistry and
microbiology.
Prior Knowledge: You must know the following rules of exponents.
1. Product Rule: xa  xb  xab . Ex. x 2  x5  x 25  x7 and 103 109  103( 9)  106.
x8
xa
10
a b

x
.
2. Quotient Rule: b
Ex. 5  x85  x3 and
 101( 6)  1016  107.
6
x
x
10
3. Reciprocal Rule: x  a 
1
1
1
1
. Ex. x 2  2 ,
 108 , and
 105.
a
8
5
x
x
10
10
Materials: You will need a scientific calculator (although some instructors may require that you
do this work without a calculator. Check with your instructor.)
Part I: Practice Exercises
Simplify the following expressions. Express your final answers without negative
exponents.
1. x  x 
2. 3m 5m 
4. 6t 5  7t 8 
5.
2
5
6
27v 4

3v 9
12 y 9

3.
8 y4
6. 7m6  8m4
3
Now, check your answers. If you have any questions, see a tutor before moving on to the
next section.
1. x 7 ; 2. 15m7 ; 3.
3 y5
42
9
56
; 4. 3 ; 5. 5 ; 6. 10
2
t
v
m
Part II. Powers of 10
Simplify the following expressions. Express your final answers as powers of 10 (the
exponents may be positive, negative or zero). See examples 1 and 2.
Example 1: 105 108  1058  103
Example 2:
102
 1026  108
106
1. 104 108 
2. 105 104 
3.
10

102
105

10
5. 109 109 
6.
1

106
4.
Now, check your answers. If you have any questions, see a tutor before moving on to the
next section.
1. 1012 ; 2. 10; 3. 103 ; 4. 106 ; 5. 100 ; 6. 106
4
Part III. Products with powers of 10
Multiplying by powers of ten is easy. You just need to know where to move the
decimal point. Remember, if the exponent of 10 is positive, then you are multiplying
by a number that is greater than 1, and your answer will get larger, so you must move
the decimal point to the right. If the exponent of 10 is negative, then you are
multiplying by a number that is less than 1, and your answer will get smaller, so you
must move the decimal point to the left.
Study the following examples, and then do the following multiplications.
Example 1: 2.53 105  253,000
Note that the decimal point in the answer is not written, but it is assumed to be after the last
zero, 5 places to the right of where it was originally.
Example 2: 2.53 105  0.0000253
Note that the decimal point in the answer is five places to the left of where it was originally.
1. 3.7 104 
2. 0.005 105 
3. 7.1106 
4. 217.4 104 
Part IV. Scientific Notation
A. Definition of Scientific Notation
Sometimes scientists need to write very big or very small numbers that contain many
zeros, such as the number of molecules of a substance in one mole of the substance,
known as Avogadro Constant, which is 602,214,179,000,000,000,000,000; or, the
radius of an electron, which is approximately 0.00000000000000281794 meters.
Imagine having to write numbers such as these several times in a lab report!! Well,
scientists use a more efficient notation known as scientific notation.
A number is in scientific notation when it is in the form A10 x where A is a number
greater than or equal to one and less than 10, and x is an integer (which means that the
exponent cannot be a fraction or a decimal). For example, the following numbers are
in scientific notation:
a) 2.8 109 ,
b) 7.032 107 ,
c) 1103 ,
d) 9.9976 10
5
Explain why each of the following numbers is NOT in scientific notation. See
examples 1 and 2.
Example 1. 12.5 104 : The number 12.5 is not less than 10
Example 2. 3.14 105.4 : The exponent of 10 is not an integer
1
1. 110 2 :
2. 36 108 :
3. 0.98 106 :
1
2
4. 110 :
5. 13.25 100.7 :
6. 2.5  610 :
7. 6.408 :
8. 4.58  104 :
6
B. Standard Notation and Scientific Notation.
One of the skills you need to develop is the ability to change numbers back and forth
between scientific notation and standard notation. Let us practice.
Write the following numbers in standard notation. See examples 3 and 4.
Example 3: 2.8 105  280,000
Example 4: 5.01108  0.0000000501
9. 1.1107 
10. 7 103 
11. 9.03 106 
12. 8.403 107 
13. 7.2 104 
14. 4.3786 105 
Sometimes you will be required to write numbers in scientific notation. When these
numbers are in standard form, there a simple 3-step process to do this:
1. Write the first digit the given number that is NOT zero and place a decimal
point after this digit.
2. Copy all the remaining digits of the given number until you get to the last digit
that is NOT zero.
3. Multiply by the appropriate power of ten which would move the decimal point
to its original position.
For example to write the number 52,300,000 in scientific notation we carry out the 3step process as follows:
Step 1: 5.
[The first digit that is not zero is 5]
Step 2: 5.23
[The last digit that is not zero is 3]
Step 3: 5.23 107
[The original position of the decimal point is 7 places to the
right of where it is now, so we multiply by 107 ]
So, 52,300,000 = 5.23 107
7
Now, you carry out the 3-step process to write the number 0.000008069 in scientific
notation. Fill in the blanks.
Step 1: _____
[The first digit that is not zero is ___ ]
Step 2: _____
[The last digit that is not zero is ___ ]
Step 3: ___________ [The original position of the decimal point is ___ places to
the _______ of where it is now, so we multiply by _____ ]
So, 0.000008069 = ________________
Now, write the following numbers in scientific notation.
15. 12,000,000,000 =
16. 0.0000005068 =
17. 0.0000000000001 =
18. 908,000,000 =
19. 0.962 =
20. 300 =
21. Avogadro Constant: 602,214,179,000,000,000,000,000 =
22. Radius of an electron: 0.00000000000000281794 meters =
8
C. Numbers in “Almost” Scientific Notation
Sometimes, numbers that are not in scientific notation look like they are written in
scientific notation. For example, the number 12.4 105 is not in scientific notation
because 12.4 in not less than 10. The question is, how do we write such a number in
scientific notation? One approach is to write the number 12.4 105 in standard form
and then change it into scientific notation:
12.4 105  1, 240,000  1.24 106.
Unfortunately, if the exponent of 10 is large, this method can be tedious. Imagine
having to write the number 12.4 1025 in scientific notation. Using the method we
used above would give us:
12.4 1025  124,000,000,000,000,000,000,000,000  1.24 1026.
Can you think of a different way to get the answer without having to write the number
in standard form? Stop reading at this point and think about this question now.
Write your thoughts on the lines below.
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
One alternative is to write the number being multiplied by a power of ten in scientific
notation, and then multiply the powers of ten using the rules for exponents we
reviewed at the beginning of this packet.
Let’s revisit the problem of writing 12.4 1025 in scientific notation.
First we write 12.4 in scientific notation: 12.4 = 1.24 101
So we have 12.4 1025  1.24 101 1025  1.24 1026.
Now apply the same technique to write the number 347.4 1055 in scientific notation.
Fill in the blanks.
First we write 347.4 in scientific notation: 347.4 = _____ 10___
So we have 347.4 1055  _______10 ___ 1055  ______10 ___.
9
Your turn! Write the following numbers in scientific notation.
23. 37.9 1012 
24. 0.45 108 
25. 12.6 1030 
26. 0.0045 1020 
27. 195 1012 
28. 195 1012 
154

106
30.
29.
154

106
Part V. Multiplying and Dividing in Scientific Notation
Some problems in science require you to multiply or divide numbers in scientific notation.
Typically, in these problems you need to write your final answer in scientific notation.
Study the following examples.
Example 1.
5.2 10  4 10   5.2  4  10
6
9
6
109   20.8 1015  2.08 101 1015  2.08 1016.
Example 2.
6.2 106 6.2 106

 10  0.775 104  7.75  101  104  7.75  105.
10
8 10
8 10
If you are unclear on any step of the previous two examples, ask a tutor for help before you
continue.
10
Do the following operations and write your answers in scientific notation. Use your
calculator as needed.
1.
3.4 10  6.110  
2.
 7.5 10  4.9 10  
3.
3.78 1021

8.4 1019
4.
9 10 9.9 10  
6.
3.996 10

2.7 109
3
5. 1.9 1012  8.1 
5
8
120
12
180
11
Part VI: Reflections
a) Why did you (or your instructor) decide that completing this activity was a valuable learning
experience?
b) Prior to completing this activity, did you struggle with content in any of your classes because you did
not understand scientific notation? Please be specific. Name the class and the topics with which you
struggled.
c) What was the most challenging part of this activity? How did you deal with this challenge?
d) Name something new you learned as a result of completing this activity that you think will help you
do better in any of your classes. Be specific.
12
13
M102.1 – Scientific Notation
__________________________________________________
PRINT STUDENT NAME
_______________________
STUDENT #
Tutor Feedback:
_________ Student completed the entire activity
_________ Student demonstrated understanding of the process during the discussion of his/her work
_________ Student was thoughtful in his reflections and expressed his/her thoughts using complete
sentences and proper grammar.
Additional Comments:
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
___________________________________________________
PRINT INSTRUCTOR/TUTOR NAME
___________________
DATE
INSTRUCTOR/TUTOR SIGNATURE
STUDENT – DO NOT FORGET TO TURN THIS SHEET IN
AT THE FRONT DESK!
You may not get credit for completing this DLA if you fail
to leave this sheet with the front desk receptionist.