Question
Answer
$$-4-5i-(-7+2i)+(13+i)\cdot(-4-5i)-(-7+2i)+13+i=-77i-24$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-4-5i-(-7+2i)+(13+i)\cdot(-4-5i)-(-7+2i)+13+i& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}-4-5i-(-7+2i)-52-65i-4i-5i^2-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-4-5i-(-7+2i)-5i^2-69i-52-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-4-5i-(-7+2i)+5-69i-52-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-4-5i-(-7+2i)-69i-47-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-4-5i+7-2i-69i-47-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-7i+3-69i-47-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-76i-44-(-7+2i)+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-76i-44+7-2i+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}-78i-37+13+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}-77i-24\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{13+i}\right) $ by each term in $ \left( -4-5i\right) $. $$ \left( \color{blue}{13+i}\right) \cdot \left( -4-5i\right) = -52-65i-4i-5i^2 $$ |
| ② | Combine like terms: $$ -52 \color{blue}{-65i} \color{blue}{-4i} -5i^2 = -5i^2 \color{blue}{-69i} -52 $$ |
| ③ | $$ -5i^2 = -5 \cdot (-1) = 5 $$ |
| ④ | Combine like terms: $$ \color{blue}{5} -69i \color{blue}{-52} = -69i \color{blue}{-47} $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -7+2i \right) = 7-2i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-4} \color{red}{-5i} + \color{blue}{7} \color{red}{-2i} = \color{red}{-7i} + \color{blue}{3} $$ |
| ⑦ | Combine like terms: $$ \color{blue}{-7i} + \color{red}{3} \color{blue}{-69i} \color{red}{-47} = \color{blue}{-76i} \color{red}{-44} $$ |
| ⑧ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -7+2i \right) = 7-2i $$ |
| ⑨ | Combine like terms: $$ \color{blue}{-76i} \color{red}{-44} + \color{red}{7} \color{blue}{-2i} = \color{blue}{-78i} \color{red}{-37} $$ |
| ⑩ | Combine like terms: $$ \color{blue}{-78i} \color{red}{-37} + \color{red}{13} + \color{blue}{i} = \color{blue}{-77i} \color{red}{-24} $$ |
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