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Number Theory. Argue Is not the square of an integer.

  1. Sep 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Argue that (17^4)*(5^10)*(3^5) is not the square of an integer.



    2. Relevant equations
    N/A?


    3. The attempt at a solution
    Do I break these up, and show that each is not a square? I'm not sure if that would be correct, but sqrt(17^4)=289 * sqrt(5^10)=3125 * sqrt(243)=15.5884 =...

    Since sqrt(243)=15.5884 and is not an integer then the above is not the square of an integer. Is this an efficient explanation?
     
  2. jcsd
  3. Sep 5, 2012 #2

    HallsofIvy

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    Well, it certainly doesn't show any understanding of the problem! Look at the exponents: [itex]\sqrt{17^4}= (17^4)^{1/2}= 17^2[/itex]. [itex]\sqrt{5^{10}}= (5^{10})^{1/2}= 5^5[/itex]. What about [itex]\sqrt{3^5}[/itex]?

    Do you see why the fact that 3, 5, and 17 are prime numbers is important?
    (Consider the same question about [itex]\sqrt{(8)(18)}[/itex].)
     
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