Solve x^5 -16=0 in complex number system. Please explain! Thanks!?

Answer 1

x^5 = 16
x = the fifth root of 16, or, since 16 = 2^4,
2^(4/5)

Answer 2

The best way is using polar form
16= 16<2kpi < means angle
x= 16^1/5 < 2kpi/5 with k= 0,1,2,3,4
you can put it then in binomial form by writing
x= 16^1/5( cos2kpi/5 + i *sin 2kpi5)

Answer 3

x^5 -16=0
x^5 = 16

x = 2^(4/5) = 2^(0.8)

r1 to r5 are the solutions
360°/5 = 72°

r1 = 2^(0.8)[cos0° + isin0°] = 1.74(1 + 0i)
= 1.74 + 0i
r2 = 2^(0.8)[cos72° + isin72°] =
= 0.54 + 1.66i
r3 = 2^(0.8)[cos144° + isin144°]
= -1.41 + 1.02i
r4 = 2^(0.8)[cos216° + isin216°]
= -1.41 - 1.02i
r5 = 2^(0.8)[cos288° + isin288°]
= 0.54 - 1.66i