1. ## Complex numbers

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

2. Originally Posted by LightEight
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 $z=a+ib \Longrightarrow z=|z|e^{i\phi}$, where $\phi$ is the argument of z, which can be calculated by $Arg(z)=Arctan\,\frac{y}{x}$ , TAKING INTO ACCOUNT the signs of x, y.

For example, $Arg(-1-\sqrt{3}i)=Arctan\,\sqrt{3}=\frac{\pi}{3}\,\,or\,\ ,\frac{4\pi}{3}$ . As both the real and the imaginary part of z are negative it is the second option.

For $u^{1001}$ use exponent rules and the fact that $e^h=e^{h+2k\pi i}\,,\,\,k\in \mathbb{Z}$

The rest is simple standard stuff that needs to be studied, understood and practiced in any decent alebra book.

Tonio