Using Scientific Notation

Date: 11/01/97 at 21:27:10
From: Puzzled
Subject: Scientific Notation

How would you solve a problem using scientific notation?  I know it's 
used to multiply large numbers to get a correct answer, but I don't 
understand how to do it.  Here's what I've tried:
                      
     4,567,839 x 5,493,711
      4.567839 x 10 to the sixth power x
      5.493711 x 10 to the sixth power =

I don't know where to go from here.  Do I multiply the number by the 
decimal by the exponent, or ignore the exponent and simply multiply?
Please give me an example.  

I also understand that you do different things with the problem 
depending upon the operation being used. Help!

Date: 11/03/97 at 09:38:22
From: Doctor Pipe
Subject: Re: Scientific Notation

Hello,

First, when I write exponents at the keyboard, I write 10^6 (for 
example). It types in quicker and takes up less space than writing 
"10 to the sixth power."

Now, the thing to remember when working with scientific notation to do 
multiplication is the law of exponents that says:

   a^m x a^n = a^(m+n)

For example, 10^6 x 10^6

             = 10^(6+6)
             = 10^12

Notice that the base has to be the same in both numbers! You can not 
apply the above law of exponents to 10^6 x 5^6 because the bases are 
different - the first number has a base of 10 and the second has a 
base of 5.

Getting back to your problem, you already know that
4,567,839 = 4.567839 x 10^6 and that 5,493,711 = 5.493711 x 10^6 .

This makes the problem (4.567839 x 10^6) x (5.493711 x 10^6) .

Applying the Associative Property of Multiplication

   a x (b x c) = (a x b) x c

and the Commutative Property of Multiplication

         a x b = b x a

we get:

(4.567839 x 5.493711) x (10^6 x 10^6)

By multiplying the two left-most factors and applying the above law of 
exponents to the right-most factors we get:

   25.094387 x 10^12

So, in a nutshell, we converted our large numbers to scientific 
notation, added together the exponents where we had common bases, and 
then multiplied the non-exponentiated numbers.

-Doctor Pipe,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/