Scientific Notation Handout
At various time in this course
you may be asked to work with either
very large or very small numbers. As a
matter of convenience these numbers are
often expressed as “power of ten
numbers” or scientific notation.
Scientific notation states how
many times a simple root number is
multiplied (positive power) or divided
(negative power) by 10 to achieve the
actual number. Thus a large number like
93,000,000 can be expressed as 9.3 x 107
and a very small number like .000825
can be expressed as 8.25 x 10-4.
Remembering a few simple rules will
make using scientific notation easy:
1. The root number is always
written as a decimal with no
more than one digit in front of
the decimal point. (The root
will always be greater than 1
less than 10.)
2. The exponent or power
number indicates the number
of places the decimal must be
moved from its place in the
actual number to achieve its
place in the root.
3. An actual number that is
greater than one will always
have a positive power in
scientific notation.
4. An actual number that is less
than one will always have a
negative power in scientific
notation.
Adding
Adding numbers in scientific notation is
accomplished in a 4 step process:
1. Change both numbers to the same
power of 10 by moving the decimal
in the root of one of the numbers to
be added. (Note it is usually best to
change the smaller to the greater of
the two powers.)
2. Add the two roots and write the
same power of 10 as given in the
problem.
3. Round the root number so that it has
the same precision (decimal place)
as the starting number with the
greater power.
4. If necessary adjust the number to
proper scientific notation.
Example:
Problem 4.5 x 102 + 1.25 x 103 =
1. Change both to the same power,
the greater power.
0.45 x 103 + 1.25 x 103
2. Add the roots 0.45 + 1.25 = 1.70
then express them x 10 to the
power 1.70 x 103
3. Round by using the Rules of
Significant Digits based on the
precision of original values.
4. In this case it is not necessary to
adjust.
Subtracting
Subtracting numbers in scientific
notation is accomplished using exactly the
same process as adding numbers. The only
differences is in step 2 where the roots are
subtracted instead of being added.
Example:
Problem: 2.1 x 103 . 4.0 x 102 =
1. 2.1 . 4.0 = 8.4
2. 103 + 102 = 105
Example:
3. Rounding is not necessary.
Problem: 2.3 x 10-2 - 1.2 x 10-3 =
4. Answer = 8.4 x 105
1.
Change both to the same power, the
greater power.
2.3 x 10-2 – 1.2 x 10-2
2.
Subtract the roots 2.3 - .1 = 2.18
then express them x 10 to the power 2.18 x
10-2
Dividing
Dividing in scientific notation is
accomplished in 4 steps very similar to
multiplying.
3.
Round to 2.2 x 10-2 because of the
Rules of Significant Digits
1. Divide the roots.
Multiplying
2. Subtract the powers.
Multiplying in scientific notation is
accomplished in 4 steps.
1. Multiply the roots.
2. Add the powers.
3. Adjust the proper scientific notation
with one digit in front of the
decimal.
4. Round the root number so that it has
the same number of digits as the
least number of digits in the
problem.
3. Round the root number so that is has
the same number of digits as the
least number of digits in the
problem.
4. Adjust to proper scientific notation
with one digit of the decimal.
Problem: 5.25 x 102 /2.0 x 10-1 =
1. 5.25 / 2.0 = 2.625
2. 102 – 10-1 = 103
3. Answer = 2.625 x 103
4. Round answer to 2.6 x 103
Practice
Writing in scientific notation
1. 52.5 x 10—
Multiplying in scientific notation
7. 6.2 x 102 ∙ 2.0 x 102 =____________
2. 100,000 =
x 10—
3. 0.82
=
x 10—
9. 7.1 x 102 * 7.1 x104 =___________
4. 0.0001 =
x 10—
Dividing in scientific notation
5. 0.0072 =
x 10—
6. 1.97 x 1012 =
7. 5.21 x 10-12 =
8. 2.37 x 10-5 =
9. 4.85 x 107 =
10. 6.02 x 105 =
Adding in scientific notation
1. 2.0 x 103 + 3.5 x 103 = ________
2. 4.7 x 104 + 3.5 x 103 = ________
3. 6.9 x 10-3 + 2.1 x 10-2 =________
Subtracting in scientific notation
4. 6.9 x 10-2 – 2.1 x 10-3=________
5. 9.3 x 103 – 7.84 x 102=________
6. 5.26 x 10-4 – 1.25 x 10-5 =______
8. 4.25 x 10-3 ∙ 4.0 x 104 =__________
10. 4.8 x 10-3 ÷ 4.0 x 104=_____________
11. 6.3 x 10-3 ÷ 2.1 x 104=_____________
12. 7.1 x 102 ÷ 7.1 x 104 = ________