“Do the Math”
Understanding Scientific Notation
Name: _______________________
On the AP exam and for all exams for this course, students are expected to utilize scientific
notation to solve mathematical applications. Often numeric data in the problems are very
large values and difficult to multiply or divide without the use of a calculator. If you know how
to use scientific notation, you can change those large values into simpler values using scientific
notation. While you won’t end up with the exact answer you may have obtained using a
calculator, you will come to the closest approximation of that answer. You may then choose
the correct answer (closest answer) from a list of possible answers on the exams. This tutorial
will help you to understand scientific notation and practice some calculations. As you work
through this problem set, it would be completely to your benefit to do so without a
calculator. Your quiz the first week of school on problems such as these will be done without a
calculator or any supporting notes.
I. Changing Decimal Numbers into Scientific Notation
Example 1: The number 100 expressed in scientific notation is 1.0 X 102 OR
written as 102 (1.0 X is assumed). That stated, when you are using scientific
notation, I want you to write it as 1.0 X 102 for our work in this course. You
will understand why as we work through problem sets.
When we changed 100 into scientific notation, the 100 became the 1.0 and the
exponent became the number of places the decimal moved to the left or 102
Example 2: The number 0.000004 expressed in scientific notation is 4.0 X 10-6
By moving the decimal 6 places to the right, the exponent became -6.
IMPORTANT!! How can you remember if the exponent becomes a larger or
smaller integer?
There are 2 options to determine this. Pick one that works best for you…
Option 1: If you move the decimal to the left to express a number in scientific
notation, the exponent becomes a larger integer (more positive).
Correspondingly, if you move the decimal to the right to express a number in
scientific notation, the exponent becomes a smaller integer (more negative).
Option 2: If the base number becomes smaller, the exponent integer becomes
larger (more positive). Conversely, if the base number becomes larger, the
exponent integer becomes smaller (more negative)
Example 1:
100 expressed in scientific notation is: 100 X 100. To express it in correct
format for scientific notation it would be written 1.0 X 102.
The base number became smaller (100 to 1.0) so the exponent integer had to
become larger representing a move of 2 decimal places (100 to 102). You
moved the decimal 2 places left so the exponent integer becomes a larger
integer. It changed from 0 to 2.
Example 2:
0.857 expressed in scientific notation is: 0.857 X 100. To express it in correct
format for scientific notation it would be written 8.57 X 10-1
The base number became larger (.857 to 8.57) so the exponent integer had to
become smaller representing a move of 1 decimal place (100 to 101). You
moved the decimal 1 place right so the exponent integer becomes a smaller
integer. It changed from 0 to -1
Do you see the pattern???
Move decimal to the right, exponent becomes smaller or more negative.
Move decimal to the left, exponent becomes larger or more positive.
Option 2: If the original number becomes smaller upon expressing it in
scientific notation e.g.
To write a number in correct scientific notation it is written showing only the
ones column followed by decimal places as in 5.73 X 103 and NOT 57.3 X 102
Sample Problem 3:
0.000467 (X 100 implied) written in Correct Scientific Notation is 4.67 X 10-4
Notice that the original number became larger (0.000467  4.67) and the
exponent became smaller or more negative (100  10-4).
Sample Problem 4:
32,000,000 written in Correct Scientific Notation is 3.2 X 107
Notice that the original number became smaller (32,000,000  3.2) and the
exponent became larger or more positive (100  107).
Your Turn! Complete the following practice problems by writing them in
Correct Scientific Notation. Remember you will not be able to use a
calculator on your first quiz!!
Original
Number
Expressed in Sci.
Note.
Original
Number
.266 =
.00173 =
15.467 =
1.572 =
.001302 =
0.068 =
178,134 =
61.500 =
5280 =
0.1 =
Expressed in Sci.
Note.
Continued on next page….
II. Changing numbers in scientific notation to Correct Scientific Notation
Sometimes you will have to change a number expressed in Scientific Notation
to Correct Scientific Notation. The same rules apply.
Sample Problem 3:
163.72 X 105 expressed as Correct Sci. Note. is 1.6372 X 107
Notice that the original number became smaller by moving the decimal 2
places (163.72  1.6372) and the exponent became larger or more positive
(105 107)
Sample Problem 4:
.0000654 X 10-8 expressed as Correct Sci. Note. is 6.54 X 10-13
Notice that the number became larger by moving the decimal 5 places
(.0000654  6.54) and the exponent became smaller or more negative 10-8
10-13).
Your Turn! Complete the following practice problems by writing them in
Correct Scientific Notation. Remember you will not be able to use a
calculator on your first quiz!!
Expressed in Sci.
Note.
Expressed in
Correct Sci. Note.
16.428 X10-4
.0695 X 105
275,691 X 10-2
.000059 X 10-3
5,881 X 10-3
Continued on next page….
III. Expressing numbers in Scientific Notation as Decimals
Sometimes you will be asked to change a number from scientific notation back
into its decimal original number.
Sample problem 5:
18.6 X 104  186,000
Notice the exponent had to become smaller (104  100) therefore the original
number had to become larger (18.6  186,000).
Don’t forget to place the commas where they are needed!!
Sample problem 6:
231.1 X 10-5  .002311
Notice the exponent had to become larger (10-5  100) therefore the original
number had to become smaller (231.1  .002311).
Your Turn! Complete the following practice problems by writing them in
decimal form. Yeah, yeah, yeah – no calculator please.
Expressed in Sci.
Note.
Expressed in
Decimal Form
16 X 105
8.73 X 10-3
235.7 X 10-8
.059 X 10-1
.00863 X 104
This tutorial will be collected the first day of class.