Answer
$z=a+bi$, $w=c+di$
Therefore, $\frac{}{zw}=\frac{}{z}\times\frac{}{w}$
Work Step by Step
$z=a+bi$, $w=c+di$, prove that $\frac{}{zw}=\frac{}{z}\times\frac{}{w}$
Evaluate $zw=(a+bi)(c+di)=ac+adi+bci+bdi^{2}$
Find the conjugate by changing the sign of the imaginary part of the complex number: $\frac{}{zw}=ac-adi-bci-bdi^{2}$ $(1)$
We have: $z=a+bi$, $w=c+di$, so:
$\frac{}{z}=a-bi$
$\frac{}{w}=c-di$
So $\frac{}{z} \times\frac{}{w}=(a-bi)(c-di)=ac-adi-bci+bdi^{2}$ $(2)$
From $(1)$ and $(2)$, we have: $\frac{}{zw}=\frac{}{z}\times\frac{}{w}$
