complex numbers help 
Given the following complex numbers:
$$
z=1+i\sqrt{3}
\qquad
w = 0.707 - 0.707i
$$
find the cartesian forms of the following expressions:
$$
z^2 \bar{w}\qquad\text{and}\qquad \frac{z^3}{w^9}
$$

The first one i found the answer to be 1.414 - 1.414i, is this correct?

If you know the polar forms of $z$ and $w$, then the calculation becomes much easier as mentioned in the comments.
Suppose that $z=r_{1}e^{i\theta_{1}}$ and $w=r_{2}e^{i\theta_{2}}$, then
$z^{2}\bar{w}=(r_{1}e^{i\theta_{1}})^{2}(r_{2}e^{-i\theta_{2}})=r_{1}^{2}r_{2}e^{i(2\theta_{1}-\theta_{2})}$ and $\frac{z^{3}}{w^9}=\frac{r_{1}^{3}}{r_{2}^{9}}e^{3i(\theta_{1}-3\theta_{2})}.$
Then you can use Euler's formula: $re^{i\theta}=r(\cos(\theta)+i\sin(\theta))$ to express your result in the form $x+iy.$
Now since $r_{1}=|z|=2$ and $\theta_{1}=\arctan(\frac{\sqrt{3}}{1})=\frac{\pi}{3}$
you have that $z=2e^{i\frac{\pi}{3}}.$ Similarly $w=r_{2}e^{-i\frac{\pi}{4}}$ where $r_{2}=|w|=\sqrt{(0.707)^2+(0.707)^2}\space(\approx 0.9998489885978).$
