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Math Help - Naturel-number

  1. #1
    Super Member dhiab's Avatar
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    Naturel-number

    Prove :

    \left( {\sqrt[3]{{45 + 29\sqrt 2 }} + \sqrt[3]{{45 - 29\sqrt 2 }}} \right) \in {\rm N}
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let x=\sqrt[3]{45+29\sqrt{2}}+\sqrt[3]{45-29\sqrt{2}}.

    Then x^3=\left(\sqrt[3]{45+29\sqrt{2}}+\sqrt[3]{45-29\sqrt{2}}\right)^3.

    For the right member I'll use the formula (a+b)^3=a^3+b^3+3ab(a+b). But a+b=x and ab=\sqrt[3]{343}=7

    Then x^3=90+21x\Rightarrow x^3-21x-90=0.

    The only real root, which is also integer is x=6.
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  3. #3
    Senior Member pacman's Avatar
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    Notice that (3 + sqrt 2)^3 = (45 + 29(sqrt 2)),

    thus x = [3 + sqrt(2)] + [3 - sqrt (2)] = 3 + 3 = 6.
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