I am stuck and need help:
Let z= -1 - sqrt (3i) and u= e^(i3pi/4)
Write z in exponential form, and u in Cartesian form
Find the complex roots of z; write answer in Cartesian form
Find u^1001; in Cartesian form
Exponential form for a complex number $\displaystyle z=a+ib \Longrightarrow z=|z|e^{i\phi}$, where $\displaystyle \phi$ is the argument of z, which can be calculated by $\displaystyle Arg(z)=Arctan\,\frac{y}{x}$ , TAKING INTO ACCOUNT the signs of x, y.
For example, $\displaystyle Arg(-1-\sqrt{3}i)=Arctan\,\sqrt{3}=\frac{\pi}{3}\,\,or\,\ ,\frac{4\pi}{3}$ . As both the real and the imaginary part of z are negative it is the second option.
For $\displaystyle u^{1001}$ use exponent rules and the fact that $\displaystyle e^h=e^{h+2k\pi i}\,,\,\,k\in \mathbb{Z}$
The rest is simple standard stuff that needs to be studied, understood and practiced in any decent alebra book.
Tonio