Let $\displaystyle z=\frac{(1-i)^2(1+\sqrt{3i})^3}{(1+i)^3}$

(a) express z in modulus-argument form

(b) express z in real-imaginary form

First Question: Do I need to expand out the top and bottom parts before doing anything else?

Edit: This is what I got for (b):

expanding the denominator out, we get (-2 + 2i)

multiplying by $\displaystyle \frac{(-2 - 2i)}{(-2 - 2i)}$ gives $\displaystyle \frac{(4i - 4)(1 + \sqrt{3i})^3}{8}$ = $\displaystyle \frac{(i-1)(1 + \sqrt{3i})^3}{2}$

expanding out $\displaystyle (1 + \sqrt{3i})^3$ gives:

$\displaystyle [1 + 2\sqrt{3i} + 3i](1 + \sqrt{3i}) = 1 + 3\sqrt{3i} + 9i + 3i\sqrt{3i}$

$\displaystyle 3i\sqrt{3i} = 3\sqrt{3i^3} = -3\sqrt{3i}$ thus:

$\displaystyle = 1 + 3\sqrt{3i} + 9i - 3\sqrt{3i} = 1 + 9i$

hence:

$\displaystyle z = \frac{(i -1)(1 + 9i)}{2}$

$\displaystyle z = \frac{(-10 -8i)}{2}$

$\displaystyle z = -5 - 4i$

correct? yes/no?

Any help for part (a) greatly appreciated!

FURTHER EDIT:

I've been through Moo's workings, adding in the cube power he missed and got the following:

(using Moo's notation of x, y, & t)

$\displaystyle x=\sqrt{2} e^{-i \tfrac \pi 4} \implies \boxed{x^{2}=2 e^{-i \tfrac \pi {2}}}$

$\displaystyle {y=2 e^{i \tfrac \pi 3}}\implies \boxed{y^{3}=8 e^{i \pi} = -8}$

$\displaystyle t=\sqrt{2} e^{i \tfrac \pi 4} \implies \boxed{t^{3}=2 \sqrt{2} e^{i \tfrac{{3}\pi} 4}}$

which gives:

$\displaystyle

z=\frac{2 e^{-i \tfrac \pi 2} \cdot -8}{2 \sqrt{2} e^{i \tfrac{3\pi} 4}} \implies

-4\sqrt{2} \cdot \frac{e^{-i \tfrac \pi 2}}{e^{i \tfrac{3\pi} 4}} \implies -4\sqrt{2} \cdot e^{i (\tfrac {-\pi} {2} - \tfrac {3\pi} {4})} \implies 4\sqrt{2} \cdot e^{-i \tfrac {5\pi} {4}}

$

$\displaystyle

z= -4\sqrt{2} (cos \tfrac {-5\pi} {4} + isin \tfrac {-5\pi} {4}) \implies-4\sqrt{2}(-\tfrac {1} {\sqrt{2}} + i \tfrac {1} {\sqrt{2}})$

$\displaystyle

z= 4 - 4i$

which is ever-so-slightly different to what I got by expanding it all out.

What the hell am I doing wrong here?! I've spent 1/2 the morning going over this and just can't for the life of me see where I've gone wrong. Any help desperately appreciated!