<s>[INST] Help with complex numbers Algebraic form of the $z^3 = (3 + i)^6$. Can someone help me to solve this? My answer is $z = 8 + 6i$, but I'm not sure that this is true.
 [/INST] Whenever you have $z^3 = k$ there will be $3$ solutions.
If $z^3 = (3+i)^6 = ((3+i)^2)^3$ then one solution is obviously $z = (3+i)^2 = 9 + 6i + i^2 = 9+6i -1 = 8+6i$.
But that is only one solution.
However $w^3 = 1$ has three solutions.  
If we let $w$ be one of the solutions and we let $z = w(8+6i) = w(3+i)^2$ then $z^3 = w^3((3+i)^2)^3 = 1*(3+i)^6 = (3+i)^6$.  
So if the three solutions to $w^3=1$ are $1, b,c$ then the three solutions to $z^3 = (3+i)^6$ are $8 + 6i, b(8+6i), c(8+6i)$.
So we nee to solve $w^3 = 1$.
Let $w = m+ni$ (where $m,n$ are real) then $(m+ni)^3 = m^3 + 3m^2ni + 3mn^2i^2 + n^3i^3=$
$m^3 + 3m^2ni -3mn^2 - n^3i = $
$(m^3 - 3mn^2) + (3m^2n-n^3)i = $
$1 + 0i$
So $m^3 -3mn^2 =1$ and $3m^2n -n^3=0$.
$3m^2n -n^3= n(3m^2-n^2) = n(m\sqrt 3 -n)(m\sqrt 3 + n) = 0$ so
either:
1)  $n = 0$ and $m^3 = 1$ so $m = 1$ and $w = 1 + 0i = 1$.
2) $m\sqrt 3 - n = 0$ and $m\sqrt 3 = n$ so $m^3 -3mn^2 = 1$ and so
$m^3 - 3m(m\sqrt 3)^2 = m^3 - 3m(3m^2) =m^3 -9m^3 = -8m^3 =1$ so $m = -\frac 12$ and $w = -\frac 12 -\frac {\sqrt 3}2 i$.
3) $m\sqrt 3+ n = 0$ so $n = -m\sqrt 3$ and so
$m^3 - 3m(-m\sqrt 3)^2=.... = 1$ so $m = -\frac 12$ and $w = -\frac 12 + \frac {\sqrt 3}2i$.
So the three solutions to $z^3 =(3+i)^6$ are:
1)$8+6i$
2)$(-\frac 12 -\frac {\sqrt 3}2 i)(8 + 6i) = -4 -4\sqrt 3i - 3i -3\sqrt 3 i^2 = (-4+3\sqrt 3) -(4\sqrt 3-3)i$.
and 3) $(-\frac 12 +\frac {\sqrt 3}2 i)(8 + 6i) = -4 +4\sqrt 3i - 3i +3\sqrt 3 i^2 = (-4-3\sqrt 3) +(4\sqrt 3-3)i$
....
However once you learn polar notation that $z = m + ni = r(\cos \theta + i\sin \theta) := re^{i\theta}$ where $r = \sqrt{m^2 + n^2}$ and $\theta$ is the solutions to $\cos \theta = \frac mr$ and $\sin \theta = \frac nr$, once you learn that these problems become MUCH easier.
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