Question #11997

find all complex numbers z such that z^4 =-1. write the answers in both polar from and the form a+ib, and sketch them in the complex plane.

Expert's answer

z = (-1)^(1/4) =

= ( cosπ + i·sinπ )^(1/4) =

= |de Moivre's formula| =

= cos((π + 2πk)/4) + i·sin((π + 2πk)/4), k = 0, 1, 2, 3.

z1 = |k = 0| = cos(π/4)& + i·sin(π/4)& = (1/√2)(1+i);

z2 = |k = 1| = cos(3π/4) + i·sin(3π/4) = (1/√2)(-1+i);

z3 = |k = 2| = cos(5π/4) + i·sin(5π/4) = (1/√2)(-1-i);

z4 = |k = 3| = cos(7π/4) + i·sin(7π/4) = (1/√2)(1-i).

= ( cosπ + i·sinπ )^(1/4) =

= |de Moivre's formula| =

= cos((π + 2πk)/4) + i·sin((π + 2πk)/4), k = 0, 1, 2, 3.

z1 = |k = 0| = cos(π/4)& + i·sin(π/4)& = (1/√2)(1+i);

z2 = |k = 1| = cos(3π/4) + i·sin(3π/4) = (1/√2)(-1+i);

z3 = |k = 2| = cos(5π/4) + i·sin(5π/4) = (1/√2)(-1-i);

z4 = |k = 3| = cos(7π/4) + i·sin(7π/4) = (1/√2)(1-i).

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