Q:

Solving a division with imaginary numbers

How can I solve the following equation:
$z/w$
When $z= 5+5i$ and$ w =2-i$

A:

When dividing complex numbers the way to do it is to multiply by the conjugate on the top on bottom, so the bottom will become real. 
$\frac{z}{w} = \frac{z\overline{w}}{w\overline{w}}$ 
In this case you have $$\frac{(5 + 5i)(2 +i)}{(2+i)(2-i)} = \frac{(5+5i)(2+i)}{5}$$ 
So now that the bottom is real I'm sure you can solve it!

A:

You first need to find $$w^{-1}=(2-i)^{-1}$$
This is $$\frac{\bar w }{|w|^2}$$
which is $$\frac{2+i}{5}$$
Then you can easily find $$zw^{-1}$$
ADD For any complex number $w=a+bi\neq 0$ we define its modulus as $$|w|=\sqrt {a^2+b^2}$$ and it's conjugate as $$\bar w =a-bi$$
Note that $$w\bar w =a^2+b^2=|w|^2$$
so we can conclude that for every $w\neq 0$, $$w^{-1}=\frac{\bar w }{|w|^2}$$

