$$x^3 = 16777216$$
$$ \begin{matrix}x_1 = 256 & x_2 = -128+128 \sqrt{ 3 }i & x_3 = -128-128 \sqrt{ 3 }i \end{matrix} $$
$ \color{blue}{ x^{3}-16777216 } $ is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.
The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $ \dfrac{p}{q} $, where p is a factor of the trailing constant and q is a factor of the leading coefficient.
The factor of the leading coefficient ( 1 ) is 1 .The factors of the constant term (-16777216) are 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 128 }{ 1 } , ~ \pm \frac{ 256 }{ 1 } , ~ \pm \frac{ 512 }{ 1 } , ~ \pm \frac{ 1024 }{ 1 } , ~ \pm \frac{ 2048 }{ 1 } , ~ \pm \frac{ 4096 }{ 1 } , ~ \pm \frac{ 8192 }{ 1 } , ~ \pm \frac{ 16384 }{ 1 } , ~ \pm \frac{ 32768 }{ 1 } , ~ \pm \frac{ 65536 }{ 1 } , ~ \pm \frac{ 131072 }{ 1 } , ~ \pm \frac{ 262144 }{ 1 } , ~ \pm \frac{ 524288 }{ 1 } , ~ \pm \frac{ 1048576 }{ 1 } , ~ \pm \frac{ 2097152 }{ 1 } , ~ \pm \frac{ 4194304 }{ 1 } , ~ \pm \frac{ 8388608 }{ 1 } , ~ \pm \frac{ 16777216 }{ 1 } ~ $$Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.
If we plug these values into the polynomial $ P(x) $, we obtain $ P(256) = 0 $.
To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $ \color{blue}{q x - p} $. In this example:
Divide $ P(x) $ with $ \color{blue}{x - 256} $
$$ \frac{ x^{3}-16777216 }{ \color{blue}{ x - 256 } } = x^{2}+256x+65536 $$Polynomial $ x^{2}+256x+65536 $ can be used to find the remaining roots.
$ \color{blue}{ x^{2}+256x+65536 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.