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anonymous

  • one year ago

Find the fourth roots of the complex number " z1 = 1 + √3 * i

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  1. anonymous
    • one year ago
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    Part I: Write z1 in polar form. Part II: Find the modulus of the roots of z1. Part III: Find the four angles that define the fourth roots of the number z1. Part IV: What are the fourth roots of the equation " z1 = 1 + √3 * i ".

  2. anonymous
    • one year ago
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    Is my answer correct? a = 1 b = sqrt(3) sqrt(a^2 + b^2) = sqrt(1 + 3) = sqrt(4) = 2 z = 2 * (1/2 + i * sqrt(3)/2) z = 2 * (cos(pi/3 + 2pi * k) + i * sin(pi/3 + 2pi * k)) z = 2 * (cos((pi/3) * (1 + 6k)) + i * sin((pi/3) * (1 + 6k))) z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6k)) + i * sin((pi/12) * (1 + 6k))) 2^(1/4) * (cos(pi/12) + i * sin(pi/12)) 2^(1/4) * (cos(7pi/12) + i * sin(7pi/12)) 2^(1/4) * (cos(13pi/12) + i * sin(13pi/12)) 2^(1/4) * (cos(19pi/12) + i * sin(19pi/12))

  3. anonymous
    • one year ago
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    looks good to me

  4. anonymous
    • one year ago
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    modulus is 2

  5. anonymous
    • one year ago
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    angle is \(\frac{\pi}{3}\)

  6. ganeshie8
    • one year ago
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    you may verify the answers by rising the roots to 4th power you should get back the z1

  7. anonymous
    • one year ago
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    and \[\frac{\pi}{3}\times \frac{1}{4}=\frac{\pi}{12}\]

  8. anonymous
    • one year ago
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    all looks swell

  9. anonymous
    • one year ago
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    Should I simplify the roots?

  10. anonymous
    • one year ago
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    @satellite73

  11. ganeshie8
    • one year ago
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    i don't think you can simplify them further

  12. anonymous
    • one year ago
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    I meant the angles.

  13. ganeshie8
    • one year ago
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    they look good the way they are now

  14. ganeshie8
    • one year ago
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    pi/12 what can you simplify here ?

  15. anonymous
    • one year ago
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    @satellite73 said it simplifies to pi/3?

  16. ganeshie8
    • one year ago
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    Nope, satellite was referring to something else

  17. anonymous
    • one year ago
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    ok

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