As you already
know, scientific notation (a form of exponential notation)
is a concise way to
express very small or very large numbers.
Consider the speed of light, 300,000,000
m/sec. When writing this value it is very easy to "lose"
one, or more, of the zeros. It is much faster and easier
to write 3.0 x 10^{8 } m/sec or 3.0 E+8
m/sec. 


Remember that correctly written scientific notation has
two components:
(1) 
a number between 1 and
10, such that
multiplied by.... 

(2) 
a power of 10. 

One of the advantages of scientific
notation is its ease of use when performing computations.
Watch the laws of exponents at work! 
To multiply two numbers expressed
in scientific notation,
simply multiply the numbers out front and add the
exponents.
Generically speaking, this process is expressed as:
(n x 10^{a}) • (m x 10^{b}) = (n ·
m) x 10^{a+b}
Example 1:
(5.1 x 10^{4}) • (2.5 x 10^{3}) = 12.75 x 10^{7}
Oops!!
This new answer is no longer in proper scientific notation.
Proper scientific notation is 1.275 x 10^{8}
NOTE:
In real life situations, answers obtained from the
multiplication (or division) of values expressed in
scientific notation may result in answers with "more
decimal accuracy" than the original values.
Regarding ACCURACY:
If values are stated to
the greatest accuracy that they are known, then the
result of multiplication (or division) with these values
cannot be determined to any better accuracy than to the
number of digits in the least accurate number.
Regarding accuracy, the answer to Example 1 would be 1.3
x 10^{8}.
On
this site, we will be finding the mathematical results
to the multiplication, or division, of scientific
notation, WITHOUT a determination of accuracy. 
To divide two numbers expressed in
scientific notation,
simply divide the numbers out front and subtract the
exponents.
Generically speaking, this process is expressed as:
Example
2:


Example 3: 

= 1.83 x 10^{2} 
Watch out for those negative exponents!!!

Example
4:

BUT, in proper scientific
notation the answer is 2.05 x 10^{3} 
Ever wonder how these numbers are
Added or Subtracted? 
To add (or subtract) two numbers
expressed in scientific notation, be sure that the exponents in each
number are the SAME. Generically speaking:
(n x 10^{a})
+ (m x 10^{a}) = (n +
m) x 10^{a}
or
(n x 10^{a})  (m x 10^{a}) = (n
 m) x 10^{a}
If the exponents are NOT the same, the
decimal of one of the numbers has to be repositioned so that its exponent is the same as
the other number being added or subtracted. Think of it as lining up
the decimals for addition or subtraction.
Example 5: (3.2 x 10^{5}) + (5.1 x 10^{4}) =
(3.2 x 10^{5}) + (0.51 x 10^{5}) = 3.71 x 10^{5}
