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Expressing complex numbers in the x + iy form

  1. Jan 29, 2015 #1
    1. The problem statement, all variables and given/known data
    ((1-i)/(sqrt2))^42
    express in x+iy form

    2. Relevant equations
    z1/z1=(r1/r2)e^(i(theta1-theta2))

    3. The attempt at a solution
    Ive found that (1-i) has r=sqrt2 so since r is sqrt2 and x=1 y=-1 so the angle is 7pi/4
    so then I have (sqrt2e^(-i7pi/4)/sqrt2)^42
    now from here is where I dont understand where to go to obtain x+iy form since it is raised to the 42 power and I dont have a e on the bottom
     
  2. jcsd
  3. Jan 29, 2015 #2

    Stephen Tashi

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    Don't change the result back to a + bi form until you have finished raising it to the 42 power. Use the formula for [itex] (r e^{i \theta})^n = ... [/itex].
     
  4. Jan 29, 2015 #3
    that formula isnt in my book
     
  5. Jan 29, 2015 #4

    Stephen Tashi

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    It works just like the formula for [itex] (ab^k)^n = ...[/itex] works for real numbers [itex] a,b,k, [/itex] and integer [itex] n [/itex].
     
  6. Jan 29, 2015 #5
    so then ((sqrt2)e^(-i7pi/4))/sqrt2)^42
    = 1/e^147ipi/2 ?? doesnt seem like thats correct
     
  7. Jan 29, 2015 #6

    Stephen Tashi

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    You didn't make a correct application of [itex] (ab^k)^n = a^n b^{kn} [/itex] For one thing, you didn't raise [itex] a = \sqrt{2} [/itex] to a power.
     
  8. Jan 29, 2015 #7

    Mark44

    Staff: Mentor

  9. Jan 30, 2015 #8

    BvU

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    Maybe it doesn't seem correct to you, but you really did it right according to me !

    (I was wondering why you didn't let the factors sqrt(2) cancel straightaway. Matter of obfuscating notation ?)
    $${1-i\over \sqrt 2} = e^{{7\over 4}\pi}$$
    Draw a unit circle, consider the x axis the real axis and the y axis the imaginary axis, and mark your ## {1-i\over \sqrt 2}##

    You should see the ##{7\over 4}\pi## (with a plus sign ! *) , the de Moivre thing, and lots more goodies. Play with squaring the number, etc.


    *) check where ##e^{-{7\over 4}\pi}## is located on that unit circle !
     
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