Question
Answer
$$(5-3i)\cdot(7+2i)\cdot(5+3i)=-33i^2+68i+205$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5-3i)\cdot(7+2i)\cdot(5+3i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(35+10i-21i-6i^2)\cdot(5+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-6i^2-11i+35)\cdot(5+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(6-11i+35)\cdot(5+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(-11i+41)\cdot(5+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-55i-33i^2+205+123i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-33i^2+68i+205\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{5-3i}\right) $ by each term in $ \left( 7+2i\right) $. $$ \left( \color{blue}{5-3i}\right) \cdot \left( 7+2i\right) = 35+10i-21i-6i^2 $$ |
| ② | Combine like terms: $$ 35+ \color{blue}{10i} \color{blue}{-21i} -6i^2 = -6i^2 \color{blue}{-11i} +35 $$ |
| ③ | $$ -6i^2 = -6 \cdot (-1) = 6 $$ |
| ④ | Combine like terms: $$ \color{blue}{6} -11i+ \color{blue}{35} = -11i+ \color{blue}{41} $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{-11i+41}\right) $ by each term in $ \left( 5+3i\right) $. $$ \left( \color{blue}{-11i+41}\right) \cdot \left( 5+3i\right) = -55i-33i^2+205+123i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-55i} -33i^2+205+ \color{blue}{123i} = -33i^2+ \color{blue}{68i} +205 $$ |
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