1. ## [SOLVED] Complex No.s

How exactly do I do these questions:

1. z = 1 + $\sqrt{3}$ . Find the smallest positive integer n for which $z^n$ is real and evaluate $z^n$ for this value of n. Show that there is no integral value of n for which $z^n$ is imaginary.

2. Find the modulus of $\frac{7-i}{3-4i}$. Evaluate $tan[\arctan\frac{4}{3} - \arctan\frac{1}{7}].$Hence, find the principal argument of $\frac{7-i}{3-4i}$ in terms of $\pi$.

2. Oh w8, found the solutions ...

SORRY!

3. Originally Posted by xwrathbringerx
1. z = 1 + $\sqrt{3}$ . Find the smallest positive integer n for which $z^n$ is real and evaluate $z^n$ for this value of n. Show that there is no integral value of n for which $z^n$ is imaginary.
I am sure that you mean $z = 1 + \sqrt 3 \color{red}~i$. In that case $\arg (z) = \frac{\pi }{3}$.
What is the smallest $n$ such that $n\left(\frac{\pi }{3}\right)=k\pi$, that is a multiple of $\pi$.

Is the any $n$ such that $n\left(\frac{\pi }{3}\right)=k\left(\frac{\pi }{2}\right)?$