Question
Answer
$$(1+i)\cdot(1-i)\cdot(2+i)=2i+4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+i)\cdot(1-i)\cdot(2+i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1-i+i-i^2)\cdot(2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-i^2+1)\cdot(2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1+1)\cdot(2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2\cdot(2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4+2i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}2i+4\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+i}\right) $ by each term in $ \left( 1-i\right) $. $$ \left( \color{blue}{1+i}\right) \cdot \left( 1-i\right) = 1 -\cancel{i}+ \cancel{i}-i^2 $$ |
| ② | Combine like terms: $$ 1 \, \color{blue}{ -\cancel{i}} \,+ \, \color{blue}{ \cancel{i}} \,-i^2 = -i^2+1 $$ |
| ③ | $$ -i^2 = -(-1) = 1 $$ |
| ④ | Combine like terms: $$ \color{blue}{1} + \color{blue}{1} = \color{blue}{2} $$ |
| ⑤ | Multiply $ \color{blue}{2} $ by $ \left( 2+i\right) $ $$ \color{blue}{2} \cdot \left( 2+i\right) = 4+2i $$ |
| ⑥ | Combine like terms: $$ 2i+4 = 2i+4 $$ |
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