1. ## 2 questions about complex numbers

Q1) Prove that for any two complex numbers $\displaystyle z_{1}$ and $\displaystyle z_{2}$ , $\displaystyle |\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}$. Is it true to write Arg$\displaystyle (z_{1}z_{2})=$ Arg$\displaystyle (z_{1})$$\displaystyle +$ Arg$\displaystyle (z_{2})$. Support your ideas geometrically.

Q2) Show that $\displaystyle z^{\frac{1}{n}}$ has $\displaystyle n$ distinct values for natural number $\displaystyle n$. What can you say about non-natural values of $\displaystyle n$? Why?

For question 1, i know that, if one the argument is negative, that equation isn't true because of essential argument thing but i don't know how to explain it at exam paper and how to support it geometrically.
For question 2, i didn't understand a thing. so don't know where to begin solving.

2. Q1. Start by writing $\displaystyle \displaystyle z_1 = x_1 + y_1i$ and $\displaystyle \displaystyle z_2 = x_2 + y_2i$.

Evaluate $\displaystyle \displaystyle \left|\frac{z_1}{z_2}\right|$ and $\displaystyle \displaystyle \frac{|z_1|}{|z_2|}$. Are they equal?

For the second part of that question, do you know how to work with the exponential polar form of complex numbers?

3. Originally Posted by Prove It
Q1. Start by writing $\displaystyle \displaystyle z_1 = x_1 + y_1i$ and $\displaystyle \displaystyle z_2 = x_2 + y_2i$.

Evaluate $\displaystyle \displaystyle \left|\frac{z_1}{z_2}\right|$ and $\displaystyle \displaystyle \frac{|z_1|}{|z_2|}$. Are they equal?
Thanks for it. i'll try.

Originally Posted by Prove It
For the second part of that question, do you know how to work with the exponential polar form of complex numbers?
Edit: i know how to work with those $\displaystyle e^{z}=e^x(cos{y}+isin{y})$ for $\displaystyle z=x+yi$ and $\displaystyle e^{i\theta}=cos{\theta}+i sin{\theta}$ about exponential thing. hope, you meant those.