Multiplying and Dividing Scientific Notation
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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 108  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

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 10a) (m x 10b) = (n m) x 10a+b


Example 1:
       (5.1 x 104) (2.5 x 103) = 12.75 x 107     Oops!!
                          This new answer is no longer in proper scientific notation.
                         
Proper scientific notation is 1.275 x 108

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 108.

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

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 103


 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 10a) + (m x 10a) = (n + m) x 10a

or

(n x 10a) - (m x 10a) = (n - m) x 10a

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 105) + (5.1 x 104) = (3.2 x 105) + (0.51 x 105) = 3.71 x 105