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Math Help - Complex number

  1. #1
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    Complex number

    What is 3^(3+ i(8pi/ln3))

    The answer is a number......

    Work done so far

    3^3 * 3^( i8pi/ln(3) ) = 27 * (3^ i8pi - 3^ln(3) ) = 27 *( 3^25.1i -3.34)


    3^(3+ i25.1) - 90.26

    I am not sure how to convert i to a number.........
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  2. #2
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    Re: Complex number

    Quote Originally Posted by mathsforumhelp View Post
    What is 3^(3+ i(8pi/ln3))
    I don't know how advanced you are.
    In complex numbers z^w=\exp(w\cdot \log(z)).
    Depending upon your level, you may want to use only the principal log.
    If so, we get \exp \left( {\ln (3)\left( {3 + \frac{{8\pi i}}{{\ln (3)}}} \right)} \right) = \exp \left( {\ln (27) + 8\pi i} \right) = 27
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  3. #3
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    Re: Complex number

    Hello, mathsforumhelp!

    \text{Evaluate: }\:X \;=\;3^{3+ \frac{8\pi}{\ln(3)}i}

    We have: . X \;=\;3^3\cdot3^{\frac{8\pi}{\ln(3)}i} \;=\;27\cdot3^{\frac{8\pi}{\ln(3)}i} .[1]

    Let y \:=\:3^{\frac{8\pi}{\ln(3)}i}

    Take logs: . \ln(y) \:=\:\ln\left(3^{\frac{8\pi}{\ln(3)}i}\right) \:=\:\frac{8\pi}{\ln(3)}i\cdot\ln(3) \quad\Rightarrow\quad \ln(y) \:=\:\8\pi i

    Hence: . y \:=\:e^{8\pi i} \:=\:(e^{i\pi})^8 \:=\:(\text{-}1)^8 \:=\:1


    Substitute into [1]: . X \:=\:27\cdot1 \:=\:27

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  4. #4
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    Re: Complex number

    Thanks guys.....
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