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Thread: algebra proof

  1. #1
    mms
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    algebra proof

    prove that $\displaystyle
    2a\sqrt {\frac{1}
    {{6 + 4\sqrt 3 }}} = a\left( {3^{1/4} - 3^{ - 1/4} } \right)

    $
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  2. #2
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    Hello, mms!

    This requires Olympic-level gymnastics . . .


    Prove that: .$\displaystyle 2a\sqrt {\frac{1}
    {6 + 4\sqrt 3 }} \:=\: a\left(3^{\frac{1}{4}} - 3^{ -\frac{1}{4}}\right)
    $
    The left side is: .$\displaystyle \frac{2a}{\sqrt{6 + 4\sqrt{3}}} $ .[1]

    Note that: .$\displaystyle 6 + 4\sqrt{3} \:=\:\sqrt{3}(2\sqrt{3}+4) \;=\;\sqrt{3}(\sqrt{3}+1)^2$

    . . . . . . Hence: .$\displaystyle \sqrt{6+4\sqrt{3}} \;=\;\sqrt[4]{3}(\sqrt{3}+1) $


    Then [1] becomes: .$\displaystyle \frac{2a}{\sqrt[4]{3}(\sqrt{3}+1)}$

    Multiply by $\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}-1}\!:\quad \frac{2a}{\sqrt[4]{3}(\sqrt{3}+1)}\cdot\frac{\sqrt{3}-1}{\sqrt{3}-1} \;=\;\frac{2a(\sqrt{3}-1)}{\sqrt[4]{3}\,(2)} \;=\;\frac{a(\sqrt{3}-1)}{\sqrt[4]{3}} $

    Therefore: .$\displaystyle \frac{a(3^{\frac{1}{2}} - 1)}{3^{\frac{1}{4}}} \;=\;a\left(3^{\frac{1}{4}} - 3^{-\frac{1}{4}}\right)$


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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, mms!

    This requires Olympic-level gymnastics . . .



    Pretty impressive. How did you determine this:
    $\displaystyle (2\sqrt{3}+4) \;=\;(\sqrt{3}+1)^2$

    It's true, but I would not have been able to figure that out, or even realize I would have to do that for this proof!
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  4. #4
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    Hello QM deFuturo
    Quote Originally Posted by QM deFuturo View Post
    Pretty impressive. How did you determine this:
    $\displaystyle (2\sqrt{3}+4) \;=\;(\sqrt{3}+1)^2$

    It's true, but I would not have been able to figure that out, or even realize I would have to do that for this proof!
    Note that:

    • $\displaystyle (a+b)^2 = a^2 + 2ab + b^2$

    and

    • $\displaystyle (\sqrt3)^2 = 3$

    Then notice that $\displaystyle 2\sqrt3+4 = 3 + 2\sqrt3 + 1 = (\sqrt3)^2 + 2\sqrt3 + 1$.

    Put $\displaystyle a = \sqrt3$ and $\displaystyle b = 1$ in the expansion of $\displaystyle (a+b)^2$, and you're there. I'm afraid this sort of cleverness only comes with years of practice. I (and I suspect Soroban also) have been doing these things since Noah was a boy.

    Grandad
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