Simplify the complex number: $\frac{(\sqrt{3} + 9i)^{181}}{(-12 + 48\sqrt{3}i)^{90}}$
I've tried a few things such as turning the top and bottom of the fraction into their trigonometric form or trying to factor out a power of 90 on the top and then simplifying from there. The problem is nothing is working for me and I can't see what the trick is. Can someone please help me with this? Thank you in advance.
EDIT: Here's what I did: $\dots = \frac{(-72 + 18\sqrt{3}i)^{90} (\sqrt{3} + 9i)}{12^{90} (-1 + 4\sqrt{3}i)^{90}} = (\frac{3}{2})^{90} (\frac{-4 + \sqrt{3}i}{-1 + 4\sqrt{3}i})^{90} (\sqrt{3} + 9i) = (\frac{3}{2})^{90} (\frac{16 + 15\sqrt{3}i}{49})^{90} (\sqrt{3} + 9i)$
I don't know what to do after this.
EDIT 2: One thing I noticed is that if we set $z = \sqrt{3} + 9i$ and $w = -12 + 48\sqrt{3}i$, then $|z| = \sqrt{84} \implies |z^2| = |z|^2 = 84$ and $|w| = 84$. Therefore the two numbers lay on the same circle with radius $84$. This could potentially be important but I'm not sure.
To compute by hand, note that $$\frac{(\sqrt{3} + 9i)^{2}}{(-12 + 48\sqrt{3}i)}=\frac{1+i\sqrt{3}}{2}=e^{i\pi/3}$$ so the $90$th power is easy. All that remains is the last factor of $(\sqrt{3} + 9i)$.