Question
Answer
$$(3+i)\cdot(2+4i)\cdot(3-i)\cdot(2-4i)=-160i^2+40$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3+i)\cdot(2+4i)\cdot(3-i)\cdot(2-4i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(6+12i+2i+4i^2)\cdot(3-i)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(4i^2+14i+6)\cdot(3-i)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-4+14i+6)\cdot(3-i)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(14i+2)\cdot(3-i)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(42i-14i^2+6-2i)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(-14i^2+40i+6)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(14+40i+6)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}(40i+20)\cdot(2-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}80i-160i^2+40-80i \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{80i}-160i^2+40 -\cancel{80i} \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}-160i^2+40\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{3+i}\right) $ by each term in $ \left( 2+4i\right) $. $$ \left( \color{blue}{3+i}\right) \cdot \left( 2+4i\right) = 6+12i+2i+4i^2 $$ |
| ② | Combine like terms: $$ 6+ \color{blue}{12i} + \color{blue}{2i} +4i^2 = 4i^2+ \color{blue}{14i} +6 $$ |
| ③ | $$ 4i^2 = 4 \cdot (-1) = -4 $$ |
| ④ | Combine like terms: $$ \color{blue}{-4} +14i+ \color{blue}{6} = 14i+ \color{blue}{2} $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{14i+2}\right) $ by each term in $ \left( 3-i\right) $. $$ \left( \color{blue}{14i+2}\right) \cdot \left( 3-i\right) = 42i-14i^2+6-2i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{42i} -14i^2+6 \color{blue}{-2i} = -14i^2+ \color{blue}{40i} +6 $$ |
| ⑦ | $$ -14i^2 = -14 \cdot (-1) = 14 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{14} +40i+ \color{blue}{6} = 40i+ \color{blue}{20} $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{40i+20}\right) $ by each term in $ \left( 2-4i\right) $. $$ \left( \color{blue}{40i+20}\right) \cdot \left( 2-4i\right) = \cancel{80i}-160i^2+40 -\cancel{80i} $$ |
| ⑩ | Combine like terms: $$ \, \color{blue}{ \cancel{80i}} \,-160i^2+40 \, \color{blue}{ -\cancel{80i}} \, = -160i^2+40 $$ |
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