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Math Help - Complex number-binomial theorem, real and imaginary

  1. #1
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    Complex number-binomial theorem, real and imaginary

    If z=a+bi, where a and b are real, use the binomial theorem to find the real and imaginary parts of z^5 and (z*)^5, (z*) stand for conjugate of z
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  2. #2
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    Hello, kingkaisai2!

    Exactly where is your difficulty?
    . . You don't know the binomial theorem?
    . . You can't handle i^3 ?
    . . You just want to check your answer?
    . . (It would be considerate to show us your answer.)


    If z=a+bi, where a and b are real, use the binomial theorem
    to find the real and imaginary parts of z^5 and (\overline{z})^5

    (a + bi)^5\;=\;a^5 + 5a^4(bi) + 10a^3(bi)^2 +  10a^2(bi)^3 + 5a(bi)^4 + (bi)^5

    . . . . . . = \;a^5 + 5a^4bi - 10a^3b^2 - 10a^2b^3i + 5ab^4 - b^5i

    . . . . . . = \;\underbrace{(a^5 - 10a^3b + 5ab^4)} + \underbrace{(5a^4b - 10a^2b^3 - b^5)}i
    . . . . . . . . . . . . . .R . . . . . . . . . . . . . I


    (a - bi)^5\;=\;a^5 + 5a^4(-bi) + 10a^3(-bi)^2 +  10a^2(-bi)^3 + 5a(-bi)^4 + (-bi)^5

    . . . . . . = \;a^5 - 5a^4bi - 10a^3b^2 + 10a^2b^3i + 5ab^4 + b^5i

    . . . . . . = \;\underbrace{(a^5 - 10a^3b + 5ab^4)} - \underbrace{(5a^4b - 10a^2b^3 - b^5)}i
    . . . . . . . . . . . . . .R . . . . . . . . . . . . . I

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