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

Question

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

 

Answer 1

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 = 315^o or -45^o.  (if you are using radians, it would be 7π/4, or -π/4), and n=2.

 

Therefore, (1-i)² = (√2)² cis(2*315^o) = 2 cis(630^o) = 2 cis(270^o) = 2(cos 270^o + i sin270^o) = 2(0 - i) = -2i.

 

I hope this helps. 

 

 

Answer 2

You have a parenthesis closed without opening:

 

   a + bi ( 1-i) ^2

 

    a + bi ( 1 -2i - 1) =

 

     a + bi ( 2i) =

 

     a- 2b