Calculating with Scientific Notation
Scientific notation is simply a method for expressing, and working with, very _______________ or very
_______________ numbers. It is a short hand method for writing numbers, and an easy method for
calculations. Numbers in scientific notation are made up of three parts: the _______________, the
_______________ and the _______________. Observe the example below:
5.67 x 105
This is the scientific notation for the standard number, 567 000. Now look at the number again, with the
three parts labeled.
5.67 = coefficient
10 = base
5 = exponent
In order for a number to be in correct scientific notation, the following conditions must be true:
1. The coefficient must be greater than or equal to _____ and less than _____.
2. The base must be _____.
3. The exponent must show the number of decimal _______________ that the decimal needs to be
moved to change the number to _______________ notation. A _______________ exponent means that
the decimal is moved to the _______________ when changing to standard notation.
Changing numbers from scientific notation to standard notation
Ex.1 Change 6.03 x 107 to standard notation.
remember, 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000
so, 6.03 x 107 = 6.03 x 10 000 000 = 60 300 000
answer = 60 300 000
Instead of finding the value of the base, we can simply move the decimal seven places to the right because
the exponent is 7.
So, 6.03 x 107 = 60 300 000
Now let us try one with a negative exponent.
Ex.2 Change 5.3 x 10-4 to standard notation.
The exponent tells us to move the decimal four places to the left.
so, 5.3 x 10-4 = 0.00053
Changing numbers from standard notation to scientific notation
Ex.1 Change 56 760 000 000 to scientific notation
Remember, the decimal is at the end of the final zero.
The decimal must be moved behind the five to ensure that the coefficient is less than 10, but greater
than or equal to one.
The coefficient will then read 5.676
The decimal will move 10 places to the left, making the exponent equal to 10.
Answer equals 5.676 x 1010
Now we try a number that is very small
Ex.2 Change 0.000000902 to scientific notation
The decimal must be moved behind the 9 to ensure a proper coefficient.
The coefficient will be 9.02
The decimal moves seven spaces to the right, making the exponent -7
Answer equals 9.02 x 10-7
Calculating with Scientific Notation
Not only does scientific notation give us a way of writing very large and very small numbers, it allows us
to easily do calculations as well. Calculators are very helpful tools, but unless you can do these
calculations without them, you can never check to see if your answers make sense. Any calculation
should be checked using your logic, so don't just assume an answer is correct. This page will explain the
rules for calculating with scientific notation.
Rule for Multiplication - When you multiply numbers with scientific notation, multiply the coefficients
together and add the exponents. The base will remain 10.
Ex 1. Multiply (3.45 x 107) x (6.25 x 105)
first rewrite the problem as: (3.45 x 6.25) x (107 x 105)
Then multiply the coefficients and add the exponents: 21.5625 x 1012
Then change to correct scientific notation and round to correct significant digits: 2.16 x 1013
NOTE - we add one to the exponent because we moved the decimal one place to the left.
Remember that correct scientific notation has a coefficient that is less than 10, but greater than or equal to
one.
Ex. 2. Multiply (2.33 x 10-6) x (8.19 x 103)
rewrite the problem as: (2.33 x 8.19) x (10-6 x 103)
Then multiply the coefficients and add the exponents: 19.0827 x 10-3
Then change to correct scientific notation and round to correct significant digits 1.91 x 10-2
Remember that -3 + 1 = -2
Rule for Division - When dividing with scientific notation, divide the coefficients and subtract the
exponents. The base will remain 10.
Ex. 1 Divide 3.5 x 108 by 6.6 x 104
rewrite the problem as:
3.5 x 108
--------6.6 x 104
Divide the coefficients and subtract the exponents to get:
0.530303 x 104
Change to correct scientific notation and round to correct significant digits to get: 5.3 x 103
Note - We subtract one from the exponent because we moved the decimal one place to the right.
Rule for Addition and Subtraction - when adding or subtracting in scientific notation, you must express
the numbers as the same power of 10. This will often involve changing the decimal place of the
coefficient.
Ex. 1 Add 3.76 x 104 and 5.5 x 102
move the decimal to change 5.5 x 102 to 0.055 x 104
add the coefficients and leave the base and exponent the same: 3.76 + 0.055 = 3.815 x 104
following the rules for rounding, our final answer is 3.815 x 104
Rounding is a little bit different because each digit shown in the original problem must be considered
significant, regardless of where it ends up in the answer.
Ex. 2 Subtract (4.8 x 105) - (9.7 x 104)
move the decimal to change 9.7 x 104 to 0.97 x 105
subtract the coefficients and leave the base and exponent the same: 4.8 - 0.97 = 3.83 x 105
round to correct number of significant digits: 3.83 x 105