1 find all the solution of the equation z3 8 abstract alge

1 find all the solution of the equation z^3 = -8 ( abstract algebra)

2 write the given complex number z = -1+i and z = -1-i in the polar form

z^3 = -8

z^3 = -8 + 0i

z^3 = 8(cos(pi + 2pi*k) + isin(pi + 2pi*k))

Cube-rooting both sides and using DeMorgan\'s law :

z = 2(cos[pi + 2pik/3] + isin[pi + 2pik]/3)

Plug in k = 0 :

2(cos(pi/3) + isin(pi/3))
2[1/2 + i*sqrt3/2)
1 + i*sqrt(3) ----> first solution

Plug in k = 1 :
2[cos(pi) + isin(pi)]
2(-1 + 0i)
-2 ---> second solution

Plug in k = 2 :
2[cos(5pi/3) + isin(5pi/3)]
2(1/2 + i*-sqrt3/2)
1 - i*sqrt(3) ---> third solution

So, solutions are :
1 + i*sqrt3
-2
1 - i*sqrt3

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z = -1 + i
z = x + iy

Comparing, x = -1, y = 1

r = sqrt(x^2 + y^2) ---> sqrt(2)
theta = atan(y/x) = 3pi/4

So, sqrt2(cos(3pi/4) + i*sin(3pi/4)) ---> first polar form

z = -1 - i
x = -1, y = -1
r = sqrt(2)
theta = 5pi/4

sqrt(2) * (cos(5pi/4) + i*sin(5pi/4)) ---> second polar form