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Math Help - complex numbers

  1. #1
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    complex numbers

    find the indicated power of the complex number (1+ i) to the 5th power
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  2. #2
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    Quote Originally Posted by scrappy View Post
    find the indicated power of the complex number (1+ i) to the 5th power
    (1+i)^5

    This can be done in a number of ways. The most direct way would probably be to expand the 1+i using the binomial expansion:
    (1+i)^5 = \left ( \begin{array}{c} 5 \\ 0 \end{array} \right )1^5i^0 + \left ( \begin{array}{c} 5 \\ 1 \end{array} \right )1^4i^1 + \left ( \begin{array}{c} 5 \\ 2 \end{array} \right )1^3i^2  + \left ( \begin{array}{c} 5 \\ 3 \end{array} \right )1^2i^3 + \left ( \begin{array}{c} 5 \\ 4 \end{array} \right )1^1i^4 + \left ( \begin{array}{c} 5 \\ 5 \end{array} \right )1^0i^5

    (1+i)^5 = 1 + 5i + 10i^2 + 10i^3 + 5i^4 + i^5

    (1+i)^5 = 1 + 5i - 10 - 10i + 5 + i

    (1+i)^5 = -4 - 4i

    -Dan
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  3. #3
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    Hello, scrappy!

    Exactly where is your difficulty?
    . . Multiply it out . . .


    Find: . (1+ i)^5

    (1 + i)^5\;=\;1^5 + 5(1^4)(i) + 10(1^3)(i^2) + 10(1^2)(i^3) + 5(1)(i^4) + i^5

    . . . . . . = \;1 + 5(1)(i) + 10(1)(-1) + 10(1)(-i) + 5(1)(1) + i

    . . . . . . = \;1 + 5i - 10 - 10i + 5 + i

    . . . . . . = \;-4 - 4i

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    If you are familiar with DeMoivre's Theorem, it's faster.

    Since 1 + i \:=\:\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)

    . . we have: . \left[\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)\right]^5\;=\;\left(\sqrt{2}\right)^5\left[\cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4}\right]

    Then: . 4\sqrt{2}\left[\text{-}\frac{1}{\sqrt{2}} + i\left(\text{-}\frac{1}{\sqrt{2}}\right) \right] \;=\;-4 - 4i

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