Question
Answer
$$(-1-2i)^4=-24i-7$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-1-2i)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}16i^4+32i^3+24i^2+8i+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}16-32i-24+8i+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-24i-7\end{aligned} $$ | |
| ① | $$ (-1-2i)^4 = (-1-2i)^2 \cdot (-1-2i)^2 $$ |
| ② | Find $ \left(-1-2i\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply 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}{ 2i }$. $$ \begin{aligned}\left(-1-2i\right)^2& \xlongequal{ S1 } \left(1+2i\right)^2 \xlongequal{ S2 } \color{blue}{1^2} +2 \cdot 1 \cdot 2i + \color{red}{\left( 2i \right)^2} = \\[1 em] & = 1+4i+4i^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{1+4i+4i^2}\right) $ by each term in $ \left( 1+4i+4i^2\right) $. $$ \left( \color{blue}{1+4i+4i^2}\right) \cdot \left( 1+4i+4i^2\right) = 1+4i+4i^2+4i+16i^2+16i^3+4i^2+16i^3+16i^4 $$ |
| ④ | Combine like terms: $$ 1+ \color{blue}{4i} + \color{red}{4i^2} + \color{blue}{4i} + \color{green}{16i^2} + \color{orange}{16i^3} + \color{green}{4i^2} + \color{orange}{16i^3} +16i^4 = \\ = 16i^4+ \color{orange}{32i^3} + \color{green}{24i^2} + \color{blue}{8i} +1 $$ |
| ⑤ | $$ 16i^4 = 16 \cdot i^2 \cdot i^2 =
16 \cdot ( - 1) \cdot ( - 1) =
16 $$ |
| ⑥ | $$ 32i^3 = 32 \cdot \color{blue}{i^2} \cdot i =
32 \cdot ( \color{blue}{-1}) \cdot i =
-32 \cdot \, i $$ |
| ⑦ | $$ 24i^2 = 24 \cdot (-1) = -24 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{-32i} + \color{blue}{8i} \color{red}{-24} + \color{green}{16} + \color{green}{1} = \color{blue}{-24i} \color{green}{-7} $$ |
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