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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