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Manipulating fractions

  1. Mar 9, 2017 #1
    1. The problem statement, all variables and given/known data
    This is something I need to show in order to solve the question I've been asked. I need to show that
    ##\sqrt{\frac{N_A}{N_D} } + \sqrt{\frac{N_D}{N_A} } = \frac{N_A + N_B}{\sqrt{N_A N_B}}##
    I know these two sides are equal because wolfram alpha says they are, and also because it works if I sub that into my proof. But I'm doing something really really really stupid I think, because I can't get there.

    2. Relevant equations


    3. The attempt at a solution
    I thought it might be easiest to square this and simplify, then square root. So
    ##(\sqrt{\frac{N_A}{N_D} } + \sqrt{\frac{N_D}{N_A} })^2 = \frac{N_A}{N_D} + 2\sqrt{\frac{N_A N_B}{N_A N_B}} + \frac{N_D}{N_A}##
    I suspect that's the step that's wrong but I don't know why. Carrying on anyway:
    ##= \frac{N_A}{N_D} + \frac{N_D}{N_A} + 2##
    Finding a common denominator:

    ##= \frac{N_A N_D + N_D N_A + 2N_A N_D}{N_A N_D}##
    Then finally square rooting again:
    ##=2##
    So that's very different to what I'm after and there is some really awful algebra mistake in there. Unfortunately I can't find it, any help would be very much appreciated! :)
     
  2. jcsd
  3. Mar 9, 2017 #2
    Oh yes, that is me being pretty stupid. Could just multiply first term by ##\frac{N_A}{N_A}## and the second term by ##\frac{N_D}{N_D}##, then:
    ##\sqrt{\frac{N_A^2}{N_A N_D}} + \sqrt{\frac{N_D^2}{N_A N_D} }= \frac{N_A}{\sqrt{N_A N_D}} + \frac{N_D}{\sqrt{N_A N_D}}##
     
  4. Mar 9, 2017 #3

    BvU

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    leads to ##N_A^2 + 2N_A N_D + N_D^2 \over N_A N_D## :rolleyes:
     
  5. Mar 9, 2017 #4
    Ah, that too! :smile: Oops! Thank you, I was never going to get that.
     
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