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Math Help - [SOLVED] How to Simplify if There are 3 Denominators Exist?

  1. #1
    Newbie adhiluhur's Avatar
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    Question [SOLVED] How to Simplify if There are 3 Denominators Exist?

    Well, it is quite easy to rationalize if there are only 1 or 2 sqrt denominators.
    For example: c/(sqrt a + sqrt b) --> (c(sqrt a - sqrt b))/(a^2 -b)

    But what if there were 3?
    for example:
    1/(sqrt 2 + sqrt 3 + sqrt 5) =?
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  2. #2
    Super Member craig's Avatar
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    I've never actually had to do one of these but here's how I'd go about it.

    Say you have \frac{1}{\sqrt{a} + \sqrt{b} + \sqrt{c}}.

    Multiply top and bottom by something like (\sqrt{a} - \sqrt{b} - \sqrt{c}). This gives you:

    \frac{(\sqrt{a} - \sqrt{b} - \sqrt{c})}{(\sqrt{a} + \sqrt{b} + \sqrt{c})(\sqrt{a} - \sqrt{b} - \sqrt{c})}

    Expanding this gives you

    \frac{(\sqrt{a} - \sqrt{b} - \sqrt{c})}{(a - b - c) - 2\sqrt{bc}}.

    Now this looks complicated but (a - b - c) is now an integer, let's call this D, you now have:

    \frac{(\sqrt{a} - \sqrt{b} - \sqrt{c})}{D - 2\sqrt{bc}}.

    I'm sure you can now rationalise this fraction?

    Hope this helps
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  3. #3
    Newbie adhiluhur's Avatar
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    Thank you Craig, so much. ^^
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