Question

Find the roots of z^3 = -27i
thanks

Answer 1

z=-3 multiply with-1sqrt-1
z= -3i is the answer of it

Answer 2

and it is imaginary

Answer 3

It's helpful to know that the powers of i form a cycle:
i^1 = i
I^2 = -1
i^3 = -i
i^4 = 1

After this, it goes back around to i. Larger powers can be divided by 4 and you can just take the remainder to simplify powers of i.

This should help with the -i part; the 27 should be straightforward enough.

Answer 4

\[z^3=-27i\]You know that \(i^2=-1\) so we can write\[z^3=27i^3\]\[z^3-27i^3=0\]\[z^3-(3i)^3=0\]\[(z-3i)(z^2+3iz+9i^2)=0\]so u have\[z-3i=0\]and a quadratic equation\[z^2+3iz-9=0\]