Question

are there solutions to x^3+27i=0 is so what are they?

Answer 1

x^3 = -27i
x = -3(i)^(1/3)

let's look at the i^(1/3)

i = 0 + 1i
changing that to polar form
r = 1
the hard part is to find the angle, since tanØ = y/x = 1/0 or undefined
but we know that tan π/2 is undefined, so
Ø = π/2 or 90°

so i = 1(cosπ/2 + i sinπ/2)

then i^(1.3)
= (1(cosπ/2 + i sinπ/2))^(1/3)
= cos (1/3)(π/2) + i sin (1/3)(π/2)) by De Moivre's Theorem

= cos π/6 + i sin π/6

so x = -3(cos π/6 + i sin π/6)

there are other answers, which can be obtained by using the fact that
tan 3π/2 or tan 5π/2 etc are also undefined

so you could write i in other polar forms , then use De Moivre's Theorem on those new ones.

Answer 2

X-27i