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Math Help - Complexed Numbers(polar form, factorrising, solving. etc)

  1. #1
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    Complexed Numbers(polar form, factorrising, solving. etc)

    I am having trouble with the following questions.

    1) Find  z^6\div w^4 for  z=-1+\sqrt{3i} and w=1+i

    2) solve over C,  z^2=8+15i

    3) Express  2\sqrt{3}+2i\div1-3i in polar form

    4) find all solutions of the equation  z^3-(\sqrt{5}-i)z^2+4z-4\sqrt{5}+4i=0

    5) Express  1+\sqrt{3i} in polar form
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  2. #2
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    1. Convert \displaystyle z and \displaystyle w to polars, use DeMoivre's Theorem. Then you can use the rule \displaystyle \frac{z_1}{z_2} = \frac{r_1}{r_2}\,\textrm{cis}\,{\left(\theta_1 - \theta_2\right)}.


    2. Convert to polar form, then use DeMoivre's Theorem to find the first solution. There will be two solutions, and the second is of the same magnitude and the two solutions are evenly spaced around a circle.


    3. Multiply top and bottom by the bottom's conjugate. Then convert to Polars.


    4. Start by testing different values of \displaystyle z which satisfy this equation. Then use the factor theorem, and long divide.


    5. Start by finding \displaystyle |1 + \sqrt{3}i| and \displaystyle arg{(1 + \sqrt{3}i)}.
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