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
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