Find all complex solutions for each equation Leave your answ

Find all complex solutions for each equation. Leave your answers in trigonometric form. 51. x^4 + 1 = 0 52. x^4 + 16 = 0 53. x^5 - i = 0

x^4 +16 =0

x^4 = -16

x= (-16)^1/4

write it in polar form: r = 16 ; angle x= pi

So, (-16) = 16e^i*pi

Now use De-movries theorem to find roots of x= (-16)^1/4

z = 16e^i*pi = 16(cospi + i*sinpi)

x = (16)^1/4[cos(pi+2kpi) + i*sin(pi+2kpi)]^1/4

= 2 [cos(pi/4+ kpi/2) +i*sin(pi/4 +kpi/2)]

4 roots are : k =0 ; x1 = 2[cospi/4 +*isinpi/4] = 2/sqrt2 + 2i/sqrt2

k= 1 ; x2 = 2[cos3pi/4 +isin3pi/4] = -2/sqrt2 + 2i/sqrt2

k =2 ; x3 = 2[cos5pi/4 +i*sin5pi/4] = -2/sqrt2 - i*2/sqrt2

k=3 ; x4 = 2[cos7pi/4 +i*sin7pi/4] = 2[ 1/sqrt2 - i*1/sqrt2]