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Thread: Pls help me with this complex number problem

  1. #1
    varun
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    Question Pls help me with this complex number problem

    If 1,w,w^2 are the cube roots of unity , then find the value of
    (1+w)(1+w^2)(1+w^3)(1+w^4)(1+w^5)
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  2. #2
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    Quote Originally Posted by topsquark View Post
    Hint: $\displaystyle w^2 = w$. (First of all, can you prove this?)

    -Dan
    You mean $\displaystyle w^3=1$?
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by math sucks View Post
    You mean $\displaystyle w^3=1$?
    Ahem!

    Um, yeah, what I wrote was kinda silly, wasn't it?

    What I was trying to remember was this:
    $\displaystyle w = a + ib \implies w^2 = a - ib$

    But thinking about it a bit further I see there is a simpler way:
    $\displaystyle w^4 = w$
    and
    $\displaystyle w^5 = w^2$

    So your expression becomes
    $\displaystyle 2(1 + w)^2(1 + w^2)^2$

    Now note that
    $\displaystyle (1 + w)^2 = w$
    and
    $\displaystyle (1 + w^2)^2 = w^2$

    -Dan
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