Question
Answer
$$(2-i)\cdot(2-i)\cdot(2-i)=4i^2-11i+6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2-i)\cdot(2-i)\cdot(2-i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(4-2i-2i+i^2)\cdot(2-i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1i^2-4i+4)\cdot(2-i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-1-4i+4)\cdot(2-i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(-4i+3)\cdot(2-i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-8i+4i^2+6-3i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}4i^2-11i+6\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{2-i}\right) $ by each term in $ \left( 2-i\right) $. $$ \left( \color{blue}{2-i}\right) \cdot \left( 2-i\right) = 4-2i-2i+i^2 $$ |
| ② | Combine like terms: $$ 4 \color{blue}{-2i} \color{blue}{-2i} +i^2 = i^2 \color{blue}{-4i} +4 $$ |
| ③ | $$ i^2 = -1 $$ |
| ④ | Combine like terms: $$ \color{blue}{-1} -4i+ \color{blue}{4} = -4i+ \color{blue}{3} $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{-4i+3}\right) $ by each term in $ \left( 2-i\right) $. $$ \left( \color{blue}{-4i+3}\right) \cdot \left( 2-i\right) = -8i+4i^2+6-3i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-8i} +4i^2+6 \color{blue}{-3i} = 4i^2 \color{blue}{-11i} +6 $$ |
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