Algebra question

Question

Given z³ = −z, z = a + bi, b < 0, and i = √(−1), find z⁵.

Answer 1

z = a+bi, z^3 = -z

what is z^5??

z^3 =-a-bi

z^5=z^3xz^2

z^2= a^2+2abi-b^2

z^3xz^2=-(a+bi)(a^2+2abi-b^2)= -(a^3+bia^2+2bia^2+2aib^2-ab^2-b^3)

z^5= b^3-a^3 -(5ba^2)i

or

z^5=z^6/z= z^3xz^3/(-a-bi)

= -zx-z/-(a+bi

=z^2/(a+bi

= (-a-bi)(-a-bi)/(a+bi

=(a+bi)(a+bi)/(a+bi)

= a+bi