Smaller vs. Larger Exponents and Numbers

Date: 03/24/2003 at 12:54:07
From: Toni
Subject: Numbers and exponents

Is a smaller number with a large exponent greater than a larger 
number with a smaller exponent? 

Using the numbers 1,2,3,4 what is the largest number you could make?
I'm thinking of 2^431 or 21^43.

Date: 03/25/2003 at 15:10:36
From: Doctor Ian
Subject: Re: Numbers and exponents

Hi Toni,

Generally, it's big exponents that give you power. Compare

   99 
  2    

to

    2
  99

However, the 'logic' you use to answer a question like this involves
understanding how exponents work, which is explained here:

   Properties of Exponents
   http://mathforum.org/library/drmath/view/57293.html 

So before reading on, make sure you understand the basic rules. 

Ready?  To compare numbers like

   431         43
  2      ?   21

one thing you can do is try to get similar exponents. For example, 

   431          10 43                   
  2     = 2 * (2  )                     

                    43                   
        = 2 * (1024)                     

Now, 1024 raised to the 43rd power is going to be a _lot_ bigger than
21 raised to the same power, right? 

Another way to do the comparison is to try to get close to powers of
10. For example, since 2^10 is close to 1000, we can approximate 

   431     10 43     ('~' means 'approximately equals')
  2    ~ (2  )

            3 43
       ~ (10 )

           129
       ~ 10

Let's see what happens with 21^43:

      43         43
    21   ~ (2*10)

            43     43
         ~ 2   * 10          Now we can use the same trick.

             10 4     43
         ~ (2  )  * 10

              3 4     43
         ~ (10 )  * 10

             12     43
         ~ 10   * 10

             55
         ~ 10  

I hope this helps. Write back if you'd like to talk more about this,
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/