How do I reduce (2i+2)/(1-i) with step-by-step please?

Question

I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the coefficients. I suspect maybe only the real part goes into calculating the amplitude but I can't be sure.

Answer 1

Try multiplying the numerator and denominator by $1+i$. This will give you $\frac{(2i+2)(1+i)}{1^2+1^2}$. Then, FOIL the numerator and note $i^2=-1$.

Answer 2

The secret is to multiply and divide by the conjugate of the denominator:

$$\frac{2i+2}{1-i} =\frac{2i+2}{1-i}\frac{1+i}{1+i}=\frac{2(1+i)(1+i)}{1-i^2}=\frac{(1+i^2)}{1+1}=1+2i+i^2=1+2-1=2i.$$

Answer 3

What about $$\frac{1+i}{1-i}=\frac{2\sqrt{2}(\sqrt{2}/2+i\sqrt{2}/2)}{\sqrt{2}(\sqrt{2}/2-i\sqrt{2}/2)} =2\frac{e^{i\frac{\pi}{4}}}{e^{-i\frac{\pi}{4}}}=2e^{i\frac{\pi}{2}}=2i?$$

Answer 4

Hint: Take an "i" out from top.