Answer
$$(3-7i)\cdot(6+8i)-(8-i)\cdot(9+6i)=-57i-4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3-7i)\cdot(6+8i)-(8-i)\cdot(9+6i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}18+24i-42i-56i^2-(72+48i-9i-6i^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-56i^2-18i+18-(-6i^2+39i+72) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}56-18i+18-(6+39i+72) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-18i+74-(39i+78) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-18i+74-39i-78 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-57i-4\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{3-7i}\right) $ by each term in $ \left( 6+8i\right) $. $$ \left( \color{blue}{3-7i}\right) \cdot \left( 6+8i\right) = 18+24i-42i-56i^2 $$Multiply each term of $ \left( \color{blue}{8-i}\right) $ by each term in $ \left( 9+6i\right) $. $$ \left( \color{blue}{8-i}\right) \cdot \left( 9+6i\right) = 72+48i-9i-6i^2 $$ |
| ② | Combine like terms: $$ 18+ \color{blue}{24i} \color{blue}{-42i} -56i^2 = -56i^2 \color{blue}{-18i} +18 $$Combine like terms: $$ 72+ \color{blue}{48i} \color{blue}{-9i} -6i^2 = -6i^2+ \color{blue}{39i} +72 $$ |
| ③ | $$ -56i^2 = -56 \cdot (-1) = 56 $$$$ -6i^2 = -6 \cdot (-1) = 6 $$ |
| ④ | Combine like terms: $$ \color{blue}{56} -18i+ \color{blue}{18} = -18i+ \color{blue}{74} $$Combine like terms: $$ \color{blue}{6} +39i+ \color{blue}{72} = 39i+ \color{blue}{78} $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 39i+78 \right) = -39i-78 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-18i} + \color{red}{74} \color{blue}{-39i} \color{red}{-78} = \color{blue}{-57i} \color{red}{-4} $$ |
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