Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the form a + bi. ) (1 − i )2 I need a lot of help on this because its not working for me
DeMoivre's Theorem states: (a+bi)n = (r cisθ)n = rn cis(nΘ)
r cisθ means r (cosΘ + isinΘ) where r = √(a2 + b2) and θ = arctan(b/a) evaluated in the correct quadrant.
So, in your example, a=1 and b=-1, so r = √2, and, since a and b are equal in magnitude, and in quadrant IV, we can find θ = 315o or -45o. (if you are using radians, it would be 7π/4, or -π/4), and n=2.
Therefore, (1-i)2 = (√2)2 cis(2*315o) = 2 cis(630o) = 2 cis(270o) = 2(cos 270o + i sin270o) = 2(0 - i) = -2i.
I hope this helps.