Find all solutions of the equation Leave answers in trigonom

Find all solutions of the equation. Leave answers in trigonometric form. x^4 + 16 = 0 squareroot 2 cis 45 degree, squareroot 2 cis 315 degree, squareroot 2 cis 135 degree, squareroot 2 cis 225 degree 16 cis 45 degree, 16 cis 315 degree, 16 cis 135 degree, 16 cis 225 degree 2 cis 45 degree, 2 cis 315 degree, 2 cis 135 degree, 2 cis 225 degree

Solution

x^4 = -16

x = (-16)^1/4

Using Demoivres theorem:

nth roots of -16 : xn = 2(cos(pi +2pik) + i*sin(pi + 2pik))^1/4

= 2(cos(pi/4 +pi*k/2) + i*sin(pi/4 +pi*k/2) )

k=0 x1 = 2[cos(pi/4) + i*sinpi/4 ]

= 2cis(45)

k =1 ; x2 = 2(cos(pi/4 +pi/2) +i*sin(pi/4+pi/2) ) = 2cis135

Option 3