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Math Help - Complex Numbers in Exponential form

  1. #1
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    Complex Numbers in Exponential form

    Hey guys, got some questions which I need help in:

    1) Express w = (10-2.(sq. root 3)i)/(1-3(sq.root 3)i) in the exponential form and hence or otherwise find the smallest +ve integer n such that w^n is a real number.

    I expressed in exponential form and got w = 2e^i(\pi /3) but I got no idea how to do the 2nd part...

    Another similar question which I need help in is this:

    2)Given that z = (sq. root 3) + i, express z in the exponential form. Hence, find the real value of k such that z^5 + kz* = 0

    Again, I only know how to express in exponential form... z = 2e^i(\pi /6)

    Please help! Thanks in advance!
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  2. #2
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    You should know that \displaystyle e^{i\pi} = -1. What power will you have to take \displaystyle w to in order to get your exponent equal to \displaystyle i\pi?
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  3. #3
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    Quote Originally Posted by Blizzardy View Post
    1) Express w = (10-2.(sq. root 3)i)/(1-3(sq.root 3)i) in the exponential form and hence or otherwise find the smallest +ve integer n such that w^n is a real number.
    I expressed in exponential form and got w = 2e^i(\pi /3) but I got no idea how to do the 2nd part...
    Clearly the answer to part 2 is 3.
    Do you see why \left( {2e^{\frac{\pi }{3}} } \right)^3  =  - 2~?
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    Many thanks guys! Yupyup I understand Q1. =) But I still can't do Q2.

    2)Given that z = (sq. root 3) + i, express z in the exponential form. Hence, find the real value of k such that z^5 + kz* = 0

    So, z = 2e^i(\pi /6)
    After substitution, I get: 2^5e^i(5pi/6) + ke^i(-pi/6) = 0
    But how do I continue?
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  5. #5
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    Quote Originally Posted by Blizzardy View Post
    Many thanks guys! So, z = 2e^i(\pi /6)
    After substitution, I get: 2^5e^i(5pi/6) + ke^i(-pi/6) = 0
    But how do I continue?
    Just notice that \exp \left( {\frac{{5\pi }}{6}} \right) =  -\exp \left( {\frac{-\pi }{6}} \right).
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