Question

Solve the equation z^4-(1+i)=0 for the complex variable z. Show Steps.

Answer 1

z^4 = 1 + i

Answer 2

change 1 + i to trig form or exponential form , first lets use exponential

Answer 3

also keep in mind 1 + i corresponds to a radius is sqrt ( 1 + 1 ) = sqrt 2 , and angle is pi/4
so z^4 = sqrt 2 e^(pi/4*i)

Answer 4

so z = ( sqrt 2 ) ^1/4 * e^ [( pi/4*i + 2pi*n) *1/4 ]

Answer 5

we can use demoivres
z^n = r [ cos (theta +2pi*k) + i sin (theta+2pi*k) ] ----> z = r^(1/n) [cos ( 1/n*theta + 2pi/k*n) + sin ( theta/n + 2pi/k*n) ]

Answer 6

and we are given a + bi , so theta = arctan b/a , and r = sqrt ( a^2 + b^2)