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Math Help - Find the value of Z^69

  1. #1
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    Exclamation Find the value of Z^69

    If z =  \frac{\sqrt{3}+1}{2} find the value of z^{69}
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  2. #2
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    Write the complex number on the form z=re^{\theta i}, and then compute z^{69}=r^{69}e^{69\theta i}.
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    Hello, zorro!

    \text{If }z =  \frac{\sqrt{3}+1}{2}, \text{ find the value of }z^{69}

    Convert z to polar form: . z \:=\:\cos\tfrac{\pi}{6} + i\sin\tfrac{\pi}{6}


    Then use DeMoivre's Theorem: . (\cos\theta + i\sin\theta)^n \;=\;\cos(n\theta) + i\sin(n\theta)

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  4. #4
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    Am I missing something here?

    I would agree with y'all if it were


    [3^(1/2)+i]/2

    But as z is not complex??????
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  5. #5
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    Quote Originally Posted by Calculus26 View Post
    Am I missing something here?

    I would agree with y'all if it were


    [3^(1/2)+i]/2

    But as z is not complex??????
    Probably a typo. Clarification might be given in a few weeks.
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  6. #6
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    Answer please

    i have tried to solve the problem but dont know what is the right answer .Could u please let me know what is the right answer .........
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  7. #7
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    Quote Originally Posted by zorro View Post
    i have tried to solve the problem but dont know what is the right answer .Could u please let me know what is the right answer .........
    Is the number z complex or real?

    Is it \frac{\sqrt{3}+{\color{red}i}}{2} or just \frac{\sqrt{3}+1}{2}?
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  8. #8
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    Quote Originally Posted by zorro View Post
    If z = \frac{\sqrt{3}+1}{2} find the value of z^{69}
    z = \frac{\sqrt{3}+1}{2} = 1.366025404 approx.

    find the value of z^{69}
     {1.366025404}^{69} = 2221547172.+decimaldigits
    How many decimals places are needed.

    or do you need the binominal expansion of this
     \frac{ \left ( \sqrt{3}+1 \right )^{69} } {2^{69}}
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  9. #9
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    Want to know the right solution

    If z=\frac{\sqrt{3}+1}{2} , find the value of z^{69}
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  10. #10
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    Okay, let me explain it to you :

    In the form you've written, it is not possible to get a nice form for the exact value !
    If it were \frac{\sqrt{3}+i}{2}, then the problem would be less meaningless.

    Consider again your problem.
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  11. #11
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    Quote Originally Posted by zorro View Post
    If z=\frac{\sqrt{3}+1}{2} , find the value of z^{69}
    There is no simple exact answer to the question you have posted. This has been said several times now.

    So either the question has a mistake in it or an approximate answer is required. If the former, then the likely question and its answer has been given in this thread. If the latter, the answer has also been given.

    I don't see how any further progress can be made here unless the OP is prepared to acknowledge and address the above issues.
    Last edited by mr fantastic; June 30th 2009 at 03:02 PM. Reason: Too many latters
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  12. #12
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    The EXACT answer:

     \frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712}
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  13. #13
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    Quote Originally Posted by aidan View Post
    The EXACT answer:
     \frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712}
    As you can see your exact solution is a bit off.

    Go to Wolfram|Alpha. Type “((sqrt[3]+1)/2)^69” exactly then click the “=” at the right-hand side of the input box.
    Attached Thumbnails Attached Thumbnails Find the value of Z^69-pf6.gif  
    Last edited by Plato; July 4th 2009 at 04:26 PM.
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  14. #14
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    Quote Originally Posted by aidan View Post
    The EXACT answer:

     \frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712}
    Quote Originally Posted by Plato View Post
    As you can see your exact solution is a bit off.

    Go to Wolfram|Alpha. Type ((sqrt[3]+1)/2)^69 exactly then click the = at the right-hand side of the input box.
    What is more interesting about aidan's answer (other than the obvious typo: \sqrt{3+1} ) is why is it wrong (at least I think this is of interest).

    Cb
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  15. #15
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    Sorry for the typo mistake

    Quote Originally Posted by Plato View Post
    Is the number z complex or real?

    Is it \frac{\sqrt{3}+{\color{red}i}}{2} or just \frac{\sqrt{3}+1}{2}?

    It is
    \frac{\sqrt{3}+i}{2}
    Now please can any one give me the right solution to this question?
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