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Question about large powers and small numerical values without using a calculator

  1. Jul 31, 2008 #1
    Hello everyone :),

    I'm sure someone will have no problem helping me with this. How do I figure out large powered numerical values without using a calculator. Kept getting questions involving 2^2007 and others. How would I be able to figure out these values without using a calculator or a lot of time on my hands? I haven't studied yet how to figure this out. Hope that makes sense, and thank you for your time to helping me with this :).
     
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  3. Jul 31, 2008 #2

    Redbelly98

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    I'm not certain I understand just what you want. It seems pretty pointless to be able to write out all the digits in the number 2^2007. So how about an approximate result where you find the nearest power of 10?

    Basically, you need to know that log(2) is approximately 0.3. Using that, we can write:

    [tex]
    2^{2007} = 10^{\log(2^{2007})}
    = 10^{2007 \cdot log(2)}
    \approx 10^{2007 \ \cdot \ 0.3}
    \approx 10^{602}
    [/tex]

    Or ... you can use the fact (familiar to those knowledgeable about computers) that 2^10 is approximately 1000 or 10^3 (It's really 1024, but we're approximating here). So:

    [tex]
    2^{2007} = 2^{10 \ \cdot \ 200.7} = (2^{10})^{200.7} \approx (10^3)^{200.7} = 10^{3 \ \cdot \ 200.7} \approx 10^{602}
    [/tex]
     
  4. Jul 31, 2008 #3

    CRGreathouse

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    Or...

    the questions may have been about modular arithmetic, like "what is the last digit of 2^2007". In that case, you look at

    2^1 = 2 (mod 10)
    2^2 = 4 (mod 10)
    2^3 = 8 (mod 10)
    2^4 = 6 (mod 10)
    2^5 = 2 (mod 10)
    . . .

    and notice that the pattern repeats.
     
  5. Aug 1, 2008 #4
    I think they would be looking for modular arithmetic since this is a number theory style proof they request for these type of questions. But, this helps me out thank you, gives me somewhere to start :).
     
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