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Find ab given its relationship to the number 432

  1. Apr 14, 2013 #1
    If a,b are positive integers and (a1/2b1/3)6 = 432, then what is the value of ab?
     
  2. jcsd
  3. Apr 14, 2013 #2

    jedishrfu

    Staff: Mentor

    Is this a problem from the SAT?

    First bring the 6 inside the a and b term to get a^6/2 * b^6/3 = 432
     
  4. Apr 14, 2013 #3
    Yes, I did that. Don't know where to go from here. Seems like I just go in circles when I try to make two equations to solve for the two unknowns.
     
  5. Apr 14, 2013 #4
    Is the only way to do this just to get a3b2=432, then find the factors of 432? I tried this and then got 16*27=432, so then a=3, b=2. But I feel like there must be a different way to do this problem...
     
  6. Apr 14, 2013 #5
    Hmmmm...... It would be good if there was another way. It seems a bit too easy.
     
  7. Apr 14, 2013 #6

    Mentallic

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    Homework Helper

    You mean b=4 :smile:

    I'm not aware of another way if there is one, and I'd imagine if there were, it'd be fairly more complicated.
     
  8. Apr 15, 2013 #7

    pwsnafu

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    Science Advisor

    We are given
    ##(a^{1/2} b^{1/3})^6 = 432##
    So
    ##a^3 b^2 = a(ab)^2 = 432##
    ##(ab)^2 = \frac{432}{a}##
    LHS is a square, so test different a.
    ##a = 2 \implies \frac{432}{a} = 216## not a square
    ##a = 3 \implies \frac{432}{a} = 144##
    144 is a square, so ab = 12.
     
  9. Apr 15, 2013 #8

    Mentallic

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    Nice!
     
  10. Apr 15, 2013 #9

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Notice that the condition "a,b are positive integers" is crucial here. If a and b were allowed to be negative, there would be more solutions. If a and b were allowed to be any real numbers there would be an infinite number of solutions.
     
  11. Apr 15, 2013 #10

    pwsnafu

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    Science Advisor

    If a and b were allowed to be negative and we are allowed to use complex algebra, then yes.
    Otherwise a1/2 is undefined.
     
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