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Math Help - How to simplify?

  1. #1
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    How to simplify?

    Question directions simply state to simplify: (hehe, simply simplify)
    (20+14sqrt(2))^(1/3) + (20-14sqrt(2))^(1/3)

    WolframAlpha solution says to express 20 + 14√(2) as a cube to equal 8 + 12√(2) + 6((√2))^2+(√(2))^3. Can someone explain how these numbers were crunched up? Is it to do with (a^3+b^3)=(a+b)(a^2-ab+b^2)?
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  2. #2
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    Re: How to simplify?

    Hey facebook.

    How do you want to simplify this? If its in terms of common factors multiplication wise then I don't think you can do that.
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  3. #3
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    Re: How to simplify?

    As far as i know this kind of thing relies a lot on luck. We can try

     \sqrt[3]{20 + 14 \sqrt{2}} = a + b \sqrt{2}

    cube both sides to get

     20 + 14 \sqrt{2} = a^3 + 6ab^2 + (3a^2b + 2b^3) \sqrt{2} \ \ \ \ \ (1)

    Now we see

     a^3 + 6ab^2 = 20

     3a^2b + 2b^3 = 14

    Luckily , we find with very little effort, a = 2 , b = 1 works so

     \sqrt[3]{20 + 14 \sqrt{2}} = a + b \sqrt{2} = 2 + 1 \sqrt{2}

    Repeating the procedure using the right hand side of $(1) , you don't have to start from scratch , we find the other cube root must satisfy

     20 - 14 \sqrt{2} = a^3 + 6ab^2 + (3a^2b + 2b^3) \sqrt{2}


     a^3 + 6ab^2 = 20

     3a^2b + 2b^3 = -14

    And again with little effort we find a = 2 , b = -1 works so

     \sqrt[3]{20 - 14 \sqrt{2}} = a + b \sqrt{2} = 2 + (-1) \sqrt{2}


    So the sum of your cube roots is actually the very humble integer 4.

     \sqrt[3]{20 + 14 \sqrt{2}} + \sqrt[3]{20 - 14 \sqrt{2}} = 2 + \sqrt{2} + 2 - \sqrt{2} = 4


    He heh...

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  4. #4
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    Re: How to simplify?

    Ahhh! I knew it had to be a guess and check method! There is no systematic way other than plugging away at numbers? There will be a huge time constraint when I have to deal with this during my exam Oh well, thank you very much!
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  5. #5
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    Re: How to simplify?

    x=\left(a+b\sqrt{d}\right)^{1/3}+\left(a-b\sqrt{d}\right)^{1/3}

    is the unique real solution of the cubic equation

    x^3-3p x - 2a =0

    where

    p=\left(a^2-b^2 d\right)^{1/3}
    Last edited by Idea; August 19th 2013 at 07:25 AM. Reason: fix the latex tags
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  6. #6
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    Re: How to simplify?

    Quote Originally Posted by Idea View Post
    x=\left(a+b\sqrt{d}\right)^{1/3}+\left(a-b\sqrt{d}\right)^{1/3}

    is the unique real solution of the cubic equation

    x^3-3p x - 2a =0

    where

    p=\left(a^2-b^2 d\right)^{1/3}
    How in the world... that is so cool! Where did you derive these formulas from? Are there similar forms for x=\left(a+b\sqrt{d}\right)^{1/2}+\left(a-b\sqrt{d}\right)^{1/2} ?
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  7. #7
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    Re: How to simplify?

    Quote Originally Posted by facebook View Post
    How in the world... that is so cool! Where did you derive these formulas from? Are there similar forms for x=\left(a+b\sqrt{d}\right)^{1/2}+\left(a-b\sqrt{d}\right)^{1/2} ?
    x=\left(a+b\sqrt{d}\right)^{1/2}+\left(a-b\sqrt{d}\right)^{1/2}

    x^2=2a+2\sqrt{a^2-b^2d}

    Example

    \left(17+12\sqrt{2}\right)^{1/2}+\left(17-12\sqrt{2}\right)^{1/2} = 6
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