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Rationalise the denominator

  1. Sep 22, 2007 #1
    not sure how to solve this. havnt been told, but would like to know, thnx

    1. The problem statement, all variables and given/known data

    rationalise the denominator of

    [tex]\frac{1}{\surd2 + \surd3 + \surd5}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    i only know how to rationalise it if its like only [tex]\frac{1}{\surd2 + \surd3}[/tex]

    hope you can help
  2. jcsd
  3. Sep 22, 2007 #2


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    Staff: Mentor

    Start with

    [tex]\frac{1}{(\sqrt2 + \sqrt3) + \sqrt5}*\frac{(\sqrt2 + \sqrt3) - \sqrt5}{(\sqrt2 + \sqrt3) - \sqrt5}[/tex]
    Last edited: Sep 22, 2007
  4. Sep 22, 2007 #3
    o rite thanks, just what i was looking for :D
  5. Sep 22, 2007 #4
    hmmm, the answer book has a different answer to me.

    ill show my working, and can you confirm ive done it right. thanks.

    [tex]\frac{1}{(\sqrt2 + \sqrt3) + \sqrt5}*\frac{1}{(\sqrt2 + \sqrt3) - \sqrt5}[/tex]

    [tex]\frac{(\surd2 + \surd3) - \surd5}{(\surd2 + \surd3)^2 - 5}[/tex]

    [tex]\frac{-\surd5}{-5\surd2 -5\surd3}[/tex]

    [tex]\frac{-\surd5(-5\surd2 + 5\surd3)}{(-5\surd2 -5\surd3)(-5\surd2 + 5\surd3)}[/tex]

    [tex]\frac{5\surd10 - 5\surd15}{50 - 75}[/tex]

    [tex]\frac{\surd10 - \surd15}{10 - 15}[/tex]

    [tex]\frac{\surd10 - \surd15}{-5}[/tex]

    right now i think that working is correct? right?

    but the answer book gives

    [tex]\frac{2\surd3 + 3\surd2 - \surd30}{12}[/tex]

    :S are they the same? or different? and/or why?

    hope you can clear it up :D

    p.s. dang that was tedious to tex all that hehe.
  6. Sep 22, 2007 #5


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    Staff: Mentor

    I corrected my earlier response. Initially I had focused on the denominator, but the numerator must equal the denominator in the second term in order to preserve the value of the initial expression.

    [tex]\frac{(\surd2 + \surd3) - \surd5}{(\surd2 + \surd3)^2 - 5}[/tex] is correct.

    Now looking at the denominator

    [tex](\sqrt2 + \sqrt3)^2 - 5[/tex] = [tex](2+3+2\sqrt2\sqrt3) - 5[/tex]

    which is just [tex]2\sqrt2\sqrt3[/tex]

    The multiply the full expression by [tex]\frac{\sqrt6}{\sqrt6}[/tex]
    Last edited: Sep 22, 2007
  7. Sep 22, 2007 #6
    o rite yeah soz, i actually accounted for that in the first one without texing it, hehe

    can you please check my workings

  8. Sep 22, 2007 #7


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    Staff: Mentor

    [tex]\frac{-\surd5}{-5\surd2 -5\surd3}[/tex] This part is not correct. I'm not sure how one manage to get this.

    See my previous post regarding the denominator.
  9. Sep 22, 2007 #8
    wooops. so how would i go from the step before to the next stage?
  10. Sep 22, 2007 #9


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    Staff: Mentor

    Starting with [tex]\frac{(\surd2 + \surd3) - \surd5}{(\surd2 + \surd3)^2 - 5}[/tex]

    take what I did with the denominator, which gives

    [tex]\frac{(\surd2 + \surd3) - \surd5}{(2+3+2\sqrt2\sqrt3) - 5}[/tex]

    = [tex]\frac{(\surd2 + \surd3) - \surd5}{2\sqrt2\sqrt3}[/tex]

    and you can take it from there.
  11. Sep 22, 2007 #10
    thanks :D all sorted
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