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Math Help - roots

  1. #1
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    roots

    Compute all the 4th roots of a=1+√3 & describe where they are located in the complex plane.
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  2. #2
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    Re: roots

    According to the formula from another problem, [r , θ]ⁿ = [rⁿ, nθ]. If [r⁴, 4θ] = [1+√3, 0], then r=\sqrt[4]{1+\sqrt{3}}. You need to find all 0 ≤ θ < 2π such that 4θ is an integer multiple of 2π.
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  3. #3
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    Re: roots

    Quote Originally Posted by franios View Post
    Compute all the 4th roots of a=1+√3 & describe where they are located in the complex plane.
    Do you mean a=1+\sqrt3~i~?
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  4. #4
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    Re: roots

    Yes
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  5. #5
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    Re: roots

    Well, the idea is the same, only now r=\sqrt[4]{|a|} and you need to find all 0 ≤ θ < 2π such that 4θ equals arg(a) up to an integer multiple of 2π
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  6. #6
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    Re: roots

    Quote Originally Posted by franios View Post
    Yes
    Then a = 2\exp \left( {\frac{{i\pi }}{6}} \right).

    The four fourth roots are {\alpha _k} = \sqrt[4]{2}\exp \left( {\frac{{i\pi }}{{24}} + \frac{{i\pi k}}{2}} \right),~k=0,1,2,3
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