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Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the form a + bi. ) (1 - i )2

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2 Answers

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. 
 
 
You have a parenthesis closed without opening:
 
   a + bi ( 1-i) ^2
 
    a + bi ( 1 -2i - 1) =
 
     a + bi ( 2i) =
 
     a- 2b     

Comments

Note: expand (1-i)2 using DeMoivre's theorem.
In that case , without using polar coordinate:
 
   ( 1 -i ) ^2 =  1- 2i + 1 = -2i
But that's NOT DeMoivre's Theorem.