Answer
$$\frac{(a+bi)(c+di)(e+fi)}{g+hi}=\frac{bdfi^3+adfi^2+bcfi^2+bdei^2+acfi+adei+bcei+ace}{g+hi}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(a+bi)(c+di)(e+fi)}{g+hi}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{(1ac+adi+bci+bdi^2)(e+fi)}{g+hi} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{bdfi^3+adfi^2+bcfi^2+bdei^2+acfi+adei+bcei+ace}{g+hi}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a+bi}\right) $ by each term in $ \left( c+di\right) $. $$ \left( \color{blue}{a+bi}\right) \cdot \left( c+di\right) = ac+adi+bci+bdi^2 $$ |
| ② | Multiply each term of $ \left( \color{blue}{ac+adi+bci+bdi^2}\right) $ by each term in $ \left( e+fi\right) $. $$ \left( \color{blue}{ac+adi+bci+bdi^2}\right) \cdot \left( e+fi\right) = ace+acfi+adei+adfi^2+bcei+bcfi^2+bdei^2+bdfi^3 $$ |
| ③ | Combine like terms: $$ ace+acfi+adei+adfi^2+bcei+bcfi^2+bdei^2+bdfi^3 = bdfi^3+adfi^2+bcfi^2+bdei^2+acfi+adei+bcei+ace $$ |
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