Question
Answer
$$(1-i)^4=-4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-i)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}i^4-4i^3+6i^2-4i+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}1+4i-6-4i+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-4\end{aligned} $$ | |
| ① | $$ (1-i)^4 = (1-i)^2 \cdot (1-i)^2 $$ |
| ② | Find $ \left(1-i\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ i }$. $$ \begin{aligned}\left(1-i\right)^2 = \color{blue}{1^2} -2 \cdot 1 \cdot i + \color{red}{i^2} = 1-2i+i^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{1-2i+i^2}\right) $ by each term in $ \left( 1-2i+i^2\right) $. $$ \left( \color{blue}{1-2i+i^2}\right) \cdot \left( 1-2i+i^2\right) = 1-2i+i^2-2i+4i^2-2i^3+i^2-2i^3+i^4 $$ |
| ④ | Combine like terms: $$ 1 \color{blue}{-2i} + \color{red}{i^2} \color{blue}{-2i} + \color{green}{4i^2} \color{orange}{-2i^3} + \color{green}{i^2} \color{orange}{-2i^3} +i^4 = i^4 \color{orange}{-4i^3} + \color{green}{6i^2} \color{blue}{-4i} +1 $$ |
| ⑤ | $$ i^4 = i^2 \cdot i^2 =
( - 1) \cdot ( - 1) =
1 $$ |
| ⑥ | $$ -4i^3 = -4 \cdot \color{blue}{i^2} \cdot i =
-4 \cdot ( \color{blue}{-1}) \cdot i =
4 \cdot \, i $$ |
| ⑦ | $$ 6i^2 = 6 \cdot (-1) = -6 $$ |
| ⑧ | Combine like terms: $$ \, \color{blue}{ \cancel{4i}} \, \, \color{blue}{ -\cancel{4i}} \, \color{green}{-6} + \color{orange}{1} + \color{orange}{1} = \color{orange}{-4} $$ |
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