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Math Help - Write in a + bi Form

  1. #1
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    Write in a + bi Form

    Write the expression in the standard form a + bi.

    (1 - sqrt{5} i)^8
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  2. #2
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    Quote Originally Posted by magentarita View Post
    Write the expression in the standard form a + bi.

    (1 - sqrt{5} i)^8
    Let z = 1 - \sqrt{5} \, i.

    You need to find z^8.

    Note that z^2 = (1 - \sqrt{5} \, i) (1 - \sqrt{5} \, i) = -4 - 2 \sqrt{5} \, i = -2 (2 + \sqrt{5} \, i).

    Therefore (z^2)^2 = z^4 = [ -2 (2 + \sqrt{5} \, i) ]^2 = 4 (2 + \sqrt{5} \, i)^2 = \, ....

    Therefore (z^4)^2 = z^8 = \, ....
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  3. #3
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    great....

    Quote Originally Posted by mr fantastic View Post
    Let z = 1 - \sqrt{5} \, i.

    You need to find z^8.

    Note that z^2 = (1 - \sqrt{5} \, i) (1 - \sqrt{5} \, i) = -4 - 2 \sqrt{5} \, i = -2 (2 + \sqrt{5} \, i).

    Therefore (z^2)^2 = z^4 = [ -2 (2 + \sqrt{5} \, i) ]^2 = 4 (2 + \sqrt{5} \, i)^2 = \, ....

    Therefore (z^4)^2 = z^8 = \, ....
    A great start....
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  4. #4
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    Quote Originally Posted by magentarita View Post
    Write the expression in the standard form a + bi. (1 - sqrt{5} i)^8
    \alpha  = \arctan \left( { - \sqrt 5 } \right)
    z = \sqrt 6 e^{i\alpha }
    z^8  = \left( {\sqrt 6 } \right)^8 e^{i8\alpha }

    YOU should know how to continue!
    If not then you have no reason to ask this question!
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    \alpha  = \arctan \left( { - \sqrt 5 } \right)
    z = \sqrt 6 e^{i\alpha }
    z^8  = \left( {\sqrt 6 } \right)^8 e^{i8\alpha }

    YOU should know how to continue!
    If not then you have no reason to ask this question!
    are students usually taught how to write and manipulate the polar form of complex numbers in precalculus?
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  6. #6
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    yes....

    Quote Originally Posted by Jhevon View Post
    are students usually taught how to write and manipulate the polar form of complex numbers in precalculus?
    It's a precalculus course in preparation for calculus 1 and an upcoming state exam.
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