Exponent Rules and Working with Numbers in Scientific Notation
The rules are:
𝑎0 = 1
;
𝑎1 = 𝑎
;
1𝑛 = 1
𝑎𝑛 ∙ 𝑎𝑚 = 𝑎𝑛+𝑚 𝑒. 𝑔. 23 ∙ 24 = 27 = 128
;
𝑎𝑛 ∙ 𝑏 𝑛 = (𝑎 ∙ 𝑏)𝑛
𝑒. 𝑔.
𝑎𝑛
= 𝑎𝑛−𝑚
𝑎𝑚
𝑒. 𝑔.
;
𝑎𝑛
𝑎 𝑛
=
(
)
𝑏𝑛
𝑏
𝑒. 𝑔. 43 /23 = (4/2)3 = 8
(𝑎𝑛 )𝑚 = 𝑎𝑛∙𝑚
𝑒. 𝑔. (23 )2 = 26 = 64
;
𝑎−𝑛 = 1⁄𝑎𝑛
𝑒. 𝑔. 2−3 = 1⁄ 3 = 0.125
2
25
= 25−3 = 22 = 4
23
32 ∙ 42 = (3 ∙ 4)2 = 144
Q. Why do we need to remember these rules?
A. We actually have, at least, three reasons:
1. To write numbers in scientific notation (covered in class).
2. Compare numbers in scientific notation (covered in class).
3. Work with numbers written in scientific notation (you are responsible for reviewing this yourself –see Appendix
1A for more information on this topic).
Q. If you will be using a calculator to work with numbers written in scientific (or standard) notation, why do you need to
know how to work with them?
A. Two words: Critical Thinking.
You must avoid writing whatever the calculator displays on the screen without thinking… In our class, before you
actually start solving a problem numerically, you have to think on what you are doing and why you are doing it; in other
words, you must have a strategy to solve problems or answer questions (see the “Problem Solving” box on page 92 of
your textbook). To double check if you are actually thinking about what you are doing, means to double check if your
answer actually makes sense (as opposed to just performing calculations).
The rule in our class is:
If your answer is outrageously wrong, you will not get points for it… Why? Because it means you do
not understand the problem or you have no idea of what to expect.
If you are solving a problem, how can you have an idea of what to expect? Well, for numerical answers, after you have
your strategy in place, there will be a sequence of calculations to perform and you have two ways to gauge what the
expected result is, and those are:
a) Whenever possible, you must perform a ballpark calculation (rough estimate or approximation), which will help
you to see if you got the right answer (see “Strategies for Success” on the Textbook, page. 30).
b) When calculations are too complex, you must perform the calculation twice with your calculator and if you get a
different answer on each, a third calculation would be necessary to corroborate the answer.
At all costs: Avoid writing just whatever the calculator displays on the screen without thinking…
Dr. Reyes
Also, from computer science we have the G.I.G.O. concept (garbage in, garbage out), which refers to the fact that flawed
or nonsense input data produces a nonsense answer –which is exactly what we are trying to avoid!
The following information will help you to perform a ballpark calculation using numbers in scientific notation; as you will
see, the idea is very similar to what you most likely did when you were a kid. Example: a ballpark estimate for 21 + 83 is
100. You expect the answer to be on the vicinity (not far) from 100 since we know 20 + 80 = 100 and those rounded
numbers are close to the actual information; therefore, you would be very surprised then, if someone tells you that:
21 + 83 = 1004.
To perform a ballpark calculation in our class we will do exactly the same as above (round numbers); however, we need
to consider a couple more details due to the presence of exponents; nevertheless, the rules for exponents covered
above will help us in this regard.
Simplifying Scientific Notation – Multiplication
Since all numbers multiply one another, we can redistribute and apply exponent rules (𝑎𝑛 ∙ 𝑎𝑚 = 𝑎𝑛+𝑚 ).
(2.6 × 105 )(9.2 × 10−13 ) = (2.6)(105 )(9.2)(10−13 ) = (2.6)(9.2)(105−13 )
Once we have a set-up, we can perform the ballpark calculation or rough estimate. For example:
(2.6)(9.2)(105−13 ) ≈ (3)(10)(10−8 ) = 30 × 10−8 = (3.0 × 101 )(10−8 ) = 3(101−8 ) = 𝟑 × 𝟏𝟎−𝟕
The actual answer is: 𝟐. 𝟑𝟗𝟐 × 𝟏𝟎−𝟕. Notice this smaller number is expected, since we rounded both numbers up for
our estimate. As a rule, the power of 10 on your estimate should be identical to the actual calculation.
Simplifying Scientific Notation – Division
𝑎𝑛
Again, we can redistribute and apply exponent rules ( 𝑚 = 𝑎𝑛−𝑚 ).
𝑎
(1.247 × 10−3 ) ÷ (2.9 × 10−2 ) = (1.247 ÷ 2.9)(10−3 ÷ 10−2 )
Perform the ballpark calculation: rough estimate. For example:
(1.247 ÷ 2.9)(10−3 ÷ 10−2 ) ≈ (1.2 ÷ 3)(10−3+2 ) = (0.4)(10−3+2 ) = 0.4 × 10−1 = 4 × 10−2
The actual answer is: 𝟒. 𝟑 × 𝟏𝟎−𝟐
Q. Estimate the answer to these calculations (by hand), and then confirm your answer using a calculator:
𝟏. (2 × 10−7 )(9.1 × 10−2 ) =
𝟐. (8.5 × 106 ) ÷ (5 × 10−4 ) =
𝟑.
(3 × 1023 )(8 × 1041 )
=
2 × 10−45
Dr. Reyes
Addition and Subtraction with numbers in scientific notation.
To add (or subtract) two (or more) numbers in scientific notation, make sure the exponents in each number are the
SAME. In general:
(𝑎 × 10𝑛 ) + (𝑏 × 10𝑛 ) = (𝑎 + 𝑏) × 10𝑛
;
(𝑎 × 10𝑛 ) − (𝑏 × 10𝑛 ) = (𝑎 − 𝑏) × 10𝑛
Most likely, this implies you will have to either make an exponent larger or smaller in order to match them all; that being
the case, keep in mind the following two scenarios:
When you make an exponent smaller, the number itself gets larger; for example:
5.14 × 105
=
51.4 × 104
=
514 × 103
5,140 × 102
=
=
51,400 × 10
=
514,000
Conversely, when you make an exponent larger, the number itself gets smaller; for example:
5.14 × 10−5 = 0.514 × 10−4 = 0.0514 × 10−3 = 0.00514 × 10−2 = 0.000514 × 10−1 = 0.0000514
Sample calculation for addition:
5
(7.2 × 10 ) + (5.1 × 10
4)
redistribute
4
= (72 × 10 ) + (5.1 × 10
4)
= (72 + 5.1) × 10 = 77.1 × 104 = 𝟕. 𝟕𝟏 × 𝟏𝟎𝟓
Sample calculation for subtraction:
(3 × 10
−2 )
−𝟑
− (2.2 × 10 ) = (3 × 10
calculate
4
redistribute
−2 )
−2
− (0.22 × 10 ) = (3 − 0.22) × 10
calculate
−2
= 𝟐. 𝟕𝟖 × 𝟏𝟎−𝟐
To show you can actually change any of the exponents present, here is the same calculation as above:
(3 × 10−2 ) − (2.2 × 10−3 ) = (30 × 10−3 ) − (2.2 × 10−3 ) = (30 − 2.2) × 10−3 = 27.8 × 10−3 = 𝟐. 𝟕𝟖 × 𝟏𝟎−𝟐
Q. Find the answers to the following calculations by hand, and then confirm the answer using your calculator:
𝟏. (6.83 × 107 ) − (2.3 × 106 ) =
𝟐. (8.5 × 10−6 ) + (5 × 10−4 ) =
𝟑. (3.2 × 105 ) +
𝟒.
(4.6 × 1048 )
=
(2.0 × 1044 )
(4.0 × 106 )(0.2)
=
(50,000)(8.0 × 10−2 )
Dr. Reyes