Math - 10th grade

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Simplify: (1-i)^3

Here's how I break it down using the formula (x-y)^3 but I just don't know how to get to the final answer. Please help.



= 1-3i+3(i)^2-(i)^3



Thank you

  • Math - 10th grade -

    so far, so good. Remember that
    i^2 = -1
    i^3 = -i
    and you have
    1 - 3i - 3 + i
    -2 - 2i

  • Math - 10th grade -

    your expansion is correct, except now you have to reduce the powers of i

    look at this pattern

    i^1 = i
    i^2 = -1
    i^3 = (i^2)i = -i
    i^4 = (i^2)(i^2) = (-1)(-1) = 1
    i^5 = (i^4)(i) = i

    so it runs:
    i , -1 , -i , +1 , i , -1 , -i , +1 , .....

    notice that if the exponent is divisble by 4 i^n - = +1
    if the exponent is even but not divisible by 4 i^n = -1

    back to yours ...
    1 - 3i + 3i^2 - i^3
    = 1 - 3i - 3 + i
    = -2 - 2i

  • Math - 10th grade -

    Please explain why i^2=-1

  • Math - 10th grade -

    by definition:
    i = √-1

    square both sides
    i^2 = (√-1)(√-1) = -1

    when you multiply √x by √x you get x
    e.g. √48√48 = 48
    √12.698√12.698 = 12.698

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