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Find the value of exponents

  1. Oct 15, 2012 #1
    Find the value of


    [itex] \sqrt{-\sqrt{3}+\sqrt{3 + 8 \sqrt{7 + 4\sqrt{3}}}} [/itex]


    the options are [tex]1[/tex] , [tex]0[/tex] , [tex]2[/tex] , [tex]3[/tex]
     
    Last edited: Oct 15, 2012
  2. jcsd
  3. Oct 15, 2012 #2
    Re: exponents

    sorry , i got so excited using latex for the first time i forgot to give my attempts

    take the value as [tex]x[/tex]

    square both sides and take [tex]-sqrt{3}[/tex] to the other side , and continue doing till simplified , but this got way complicated than i intended.
     
  4. Oct 15, 2012 #3

    cepheid

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    Re: exponents

    It's not clear what the problem is asking you to do here. The "value" of the expression is just what is given.
     
  5. Oct 15, 2012 #4
    Re: exponents

    oh , sorry again , the options are 1 , 0 , 2 and 3
     
  6. Oct 15, 2012 #5
    Re: exponents

    Start by writing 7+4√3 as 4+4√3+3 which can be simplified to (2+√3)^2.
     
  7. Oct 15, 2012 #6

    Mentallic

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    Re: exponents

    Well just work backwards from the most inner radical and take approximations.

    For the answer to be equal to 0, we need to have

    [tex]\sqrt{-\sqrt{3}+\sqrt{3}}[/tex]

    which we clearly don't. For 1 we need

    [tex]\sqrt{-\sqrt{3}+(1+\sqrt{3})}[/tex]

    And using the approximation of [itex]\sqrt{3}\approx1.7[/itex] would suffice.

    For 2 we need

    [tex]\sqrt{-\sqrt{3}+(4+\sqrt{3})}[/tex]

    And finally for 3 we need

    [tex]\sqrt{-\sqrt{3}+(9+\sqrt{3})}[/tex]

    So what is the radical

    [tex]\sqrt{3+8\sqrt{7+4\sqrt{3}}}[/tex] closest to? 2.7, 5.7 or 10.7?
     
  8. Oct 15, 2012 #7

    Ray Vickson

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    Re: exponents

    Although I find it hard to believe, the original expression actually does come out exactly to a small integer value.

    RGV
     
  9. Oct 15, 2012 #8

    Mentallic

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    Re: exponents

    The surds inside surds quickly lose their value! :smile:

    What I find even more amazing is infinitely nested surds such as

    [tex]\sqrt{10+\sqrt{10+\sqrt{10...}}}=\frac{1+\sqrt{41}}{2}\approx 3.7[/tex]

    Which is a lot smaller than you'd initially guess!
     
  10. Oct 15, 2012 #9

    ehild

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    Re: exponents

    Ingenious Pranav! :cool: And the same method can be applied again to get a small integer as result.

    ehild
     
  11. Oct 15, 2012 #10

    SammyS

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    Re: exponents

    Yes, Pranav-Arora !

    I'm glad to see you figured it out before I saw this thread and racked my brain over this. (Of course, then I racked my brain over whether it's racked or wracked .)
     
  12. Oct 15, 2012 #11
    Re: exponents

    Thanks ehild and SammyS! :blushing:
     
  13. Oct 16, 2012 #12
    Re: exponents

    Awesome solution pranav , can't believe i missed that. i got the answer 2 , thank you. btw which city are you from ?
     
    Last edited: Oct 16, 2012
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