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

Answer 1

DeMoivre's Theorem states that for a complex number in polar form, z = r(cosθ+isinθ), zⁿ = rⁿ(cosnθ+isinnθ).

 

So for the complex number z = (1-i)², you must:

 

1)  First convert it from standard form, z=a+bi, to polar form, z = r(cosθ+isinθ)

 

r² = a²+b²

θ = tan^-1(b/a)

 

2)  Apply DeMoivre's Theorem (above) to find z² in polar form.

 

3)  Convert z² from polar form back to standard form (a+bi):

 

a = rⁿ*cos(nθ)

b = rⁿ*sin(nθ)