(1 - i)^7

cos(pi/4) = 1/√2   sin(pi/4) = 1/√2

1 - i = √2(cos(pi/4) - i*sin(pi/4))

(1 - i)⁷ = (√2)⁷(cos(pi/4) - i*sin(pi/4))⁷

De Moivre's theorem says  (cos(x) - i*sin(x))ⁿ = cos(nx) - i*sin(nx) so:

(1 - i)⁷ = (√2)⁷(cos(pi/4) - i*sin(pi/4))⁷  = 8√2*(cos(7pi/4) - i*sin(7pi/4)) = 8√2*(1/√2 - i*(-1/√2))

or (1 - i)⁷ =  8(1 + i)

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thx a lot :)