what is a squre root of al 1-i

Question

Question from complex no. and quadratic equations

Answer 1

Write 1 - i in exponential form and then take the square root.

 

z = ρe^iφ

 

√z = (√ρ) e^iφ/2

 

ρ = √(x² + y²) = √[(1)² + (-1)²] = √2

 

φ = tan^-1(y/x) = tan^-1(-1/1) = 7π/4

 

1 - i = √2 e^i7π/4

 

√(1 - i) = [√(√2)] e^i7π/8

 

√(1 - i) = 2^1/4 e^i7π/8 = 2^1/4 [cos(7π/8) + i sin(7π/8)]

 

You can simplify it further if you wish.