Arranging Numbers by Order of Magnitude

Date: 3/25/96 at 17:40:22
From: Jason Lee
Subject: Hard/Easy Problem?

I have this problem :

Starting with the smallest, arrange the following numbers in increasing
order of magnitude.

6^200     5^300   4^400    3^500   2^600

Here's what I did, but am unsure of:

I know 6^200 = 36^100, because using another example, let's say 2^8, 
that's the same as 4^4 or 64 = 64.

But when I try to do that with 5^300, I see I can't divide by 2 on the
exponent and square the 5, because the 100 and 150 aren't the same 
number.

So can I divide by three and cube the 5 and keep doing that and does 
that work?

Thank you for your help.

- Jason Lee

Date: 3/30/96 at 9:48:6
From: Doctor Syd
Subject: Re: Hard/Easy Problem?

Dear Jason,

Hello!  To compare the 5 given numbers, you are using a good strategy -
you are writing them in different forms, which will be easier to compare. 

Now, your strategy for writing the first number as 36^100 is excellent.  
In what form, then, could you write the other numbers so as to make the 
comparison easy?  If you have lots of numbers that are all to the same
power it is easy to see what order they go in, right?  So, let's write 
the other numbers as powers of 100 as well, and then we can easily 
compare.  

So, what you alluded to in your second to last sentence will work 
very well.  Instead of writing 5^300 as 25^150 (this isn't so helpful
in comparing it with 36^100), write 5^300 in terms of something 
that is to the 100th power, so, write 5^300 = 125^100.  You can write 
all the other numbers in terms of powers of 100, and you should get 
your answer!  

It is pretty amazing how these powers increase the numbers 
fast, eh?  Well, I hope this helps!  Write back if you have any 
questions. 

-Doctor Syd,  The Math Forum