complex numbers (1 Viewer)

[EvoRevolution]

Find all the roots of z^5+1=0?

i cant seem to get it to work.

(b) if w is the root with the least positive argument show that the roots are -1,w,-w^2,w^3 and -w^4.

 

[Trebla]

Find all the roots of z^5+1=0?

i cant seem to get it to work.

(b) if w is the root with the least positive argument show that the roots are -1,w,-w^2,w^3 and -w^4.

z⁵= -1
Let z = rcis θ
r⁵cis 5θ = cis (π + 2kπ) for some integer k
r = 1
θ = (π + 2kπ)/5
Taking k = 0, 1, 2, 3 and 4 we have the solutions:
z = cis (π/5), cis (3π/5), cis (π), cis (7π/5), cis (9π/5)

If w has the smallest positive argument then w = cis (π/5)
Then rewriting the solutions as:
z = w, w³, -1 , w⁷, w⁹
But since w satisfies z⁵ = -1, then w⁵ = -1
=> w⁷ = w⁵w² = - w²
and w⁹ = w⁵w⁴ = - w⁴

Therefore solutions are z = w, w³, -1 , - w², - w⁴

 

[EvoRevolution]

ohh i get it thanks, there was that trick to it at the end, from the question
z^5 = -1 thanks