Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the form a + bi.) (1 + i)^7

Answer 1

r = l 1 + i l = √((1)² + (1)²) = √2 θ = Tan^-1(1/1) = π/4

1 + i = r(cosθ + isinθ) = √2(cos(π/4) + isin(π/4))

By DeMoivre's Theorem,

(1 + i)⁷ = (√2)⁷[cos(7π/4) + isin(7π/4)] = 8√2[ (1/√2) - (1/√2)i ] = 8 - 8i

Answer 2

(1+i)⁷= a +bi ,a=8,b=-8

= 8-8i

= 8(1-i)

(1+i)²= 1^2+2i+i^2= 2i

(1+i)⁴= (2i)^2 =-4

(1+i)⁶= 2i(-4) = -8i

(1+i)⁷=-8i(1+i) = 8-8i