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anonymous

  • one year ago

Use DeMoivre's Theorem to find the indicated power of the following complex number. (-3 + 3i)⁴ My work r = √18 and tan θ = -1 so θ = 135° because the complex number is in Quadrant II. √18 [cos(135°) + i sin(135°)] (√18)⁴ [cos(4 × 135°) + i sin(4 × 135°)] 324[cos(540°) + i sin(540°)] 324(-1 + i × 0) 324(-1) -324

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  1. anonymous
    • one year ago
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    I just need to know if I am correct

  2. zepdrix
    • one year ago
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    Hmm I took a very different approach, got the same result though. Yay good job, looks correct! :)

  3. zepdrix
    • one year ago
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    https://www.wolframalpha.com/input/?i=%28-3%2B3i%29%5E4 Just in case you need to verify ^

  4. anonymous
    • one year ago
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    If you dont mind, can you show me how you did it?

  5. anonymous
    • one year ago
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    either way thanks a lot :)

  6. zepdrix
    • one year ago
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    My method is a little strange. I factor a 3 out to start.\[\large\rm 3^4(-1+i)^4\]I recognize that the magnitude of the real and imaginary parts are the same, that only happens at the pi/4 angles. So we need to turn these into sqrt(2)/2's. So I'll factor a sqrt(2) out of each term.\[\large\rm 3^4(\sqrt{2})^4\left(-\frac{\sqrt2}{2}+\frac{\sqrt2}{2}i\right)^4\]And as you said, this is in quadrant 2, so it's 3pi/4.\[\large\rm 3^4(\sqrt{2})^4\left(\cos\frac{3\pi}{4}+i \sin\frac{3\pi}{4}\right)^4\]Then, De'Moivre gives us,\[\large\rm 3^4(\sqrt{2})^4\left(\cos3\pi+i \sin3\pi\right)\]Sine is 0 at the 3pi angle, cosine -1,\[\large\rm 3^4(\sqrt{2})^4\left(-1+0\right)\]And then just simplify a little further to get the same result. I know, I know, my method is a little goofy :) But I like it lol

  7. anonymous
    • one year ago
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    Hahaha thanks a lot man, whatever gets the job done :P

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