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Simplifying surds upon surds

  1. Jan 23, 2009 #1

    Mentallic

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    For some surds inside of surds, they can be converted into a more simple form:

    [tex]\sqrt{a+b\sqrt{c}}=e+f\sqrt{g}[/tex]

    Such as: [tex]\sqrt{11-6\sqrt{2}}=3-\sqrt{2}[/tex]

    However, there are some that cannot be simplified into this form (as far as I know).

    Such as: [tex]\sqrt{3+\sqrt{7}}[/tex]

    I am curious to know if there is fast method in realizing whether these types of equations can be simplified.
    My only way of knowing so far is as follows:

    To see if [tex]\sqrt{31+12\sqrt{3}}[/tex] can be simplified, first I let it be equal to some general simplified form:

    [tex]\sqrt{31+12\sqrt{3}}=a+b\sqrt{3}[/tex]

    squaring both sides:

    [tex]a^2+3b^2+2\sqrt{3}ab=31+12\sqrt{3}[/tex]

    Therefore, [tex]a^2+3b^2=31[/tex] (1) and
    [tex]2\sqrt{3}ab=12\sqrt{3}[/tex] (2)

    Making a or b the subject in (2)
    [tex]b=\frac{6}{a}[/tex]

    Substituting into (1)

    [tex]a^2+3(\frac{36}{a^2})=31[/tex]

    [tex]a^4-31a^2+108=0[/tex]

    Now we have a quadratic in [tex]a^2[/tex]. I will now know from the quadratic formula if this expression can be simplified or not by looking at the discriminant. If it is a perfect square, then it can be simplified, else, it cannot be.
     
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  3. Jan 26, 2009 #2

    Gib Z

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  4. Feb 2, 2009 #3

    Mentallic

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    Thanks for the link :smile:

    For simplicities sake, I have concluded from reading through the site that to quickly check if such questions as I have posed can be denested (this is the term used), I can shorten the procedure by checking:

    Given [tex]\sqrt{A \pm B}[/tex] where [tex]A,B[/tex] all reals, [tex]A+B>0[/tex]

    Checking to see if [tex]\sqrt{A^2-B^2}[/tex] is a rational number will give me the indication whether to pursue the simplified answer.

    Of course there is a wider array of problems, but I am happy with the progress made for the moment.
     
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