Given that $z = i + i^{2016} + i^{2017}$, find $|z^{10}|$. I am told that 
$$z = i + i ^ {2016} + i ^ {2017}$$
and I have to find $|z^{10}|$. This is what I tried:
$$i ^ {2016} = (i^4)^{504} = 1 ^ {504} = 1$$
$$i ^ {2017} = i \cdot (i^4)^{504} = i \cdot 1 = i$$
So we have that:
$$z = 1 + 2i$$
And then I worked towards finding $z ^ {10}$.
$$z = 1 + 2i$$
$$z^2 = 1 + 4i + 4i^2 = -3 + 4i$$
$$z^4 = ... = -7 - 24 i$$
$$z^5 = z^4 \cdot z = (-7-24i)(1 + 2i) = ... = 41 - 38i$$
$$z^{10} = (z^5)^2 = (41 - 38i)^2 = ... = 237 -3116i $$
So we then have:
$$|z^ {10} | = \sqrt{237^2 + 3116^2} = \sqrt{56169 + 964656} = \sqrt{1020825} = 15\sqrt{4537}$$
Is this correct? Is there a better way to solve this? More efficient/faster way?

$z=1+2i$; $\overline{z}=1-2i$;
$|z|^2=z\cdot \overline{z}=1-4i^2=5$;
$|z|^{10}=(|z|^2)^5=5^5$.
