Question
Answer
$$3(-1+2i)^4+12(-1+2i)^3+39(-1+2i)^2=-108i-6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3(-1+2i)^4+12(-1+2i)^3+39(-1+2i)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3(16i^4-32i^3+24i^2-8i+1)+12(8i^3-12i^2+6i-1)+39(1-4i+4i^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}3(16+32i-24-8i+1)+12(-8i+12+6i-1)+39(1-4i-4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}3(24i-7)+12(-2i+11)+39(-4i-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}72i-21-24i+132-156i-117 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}48i+111-156i-117 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}-108i-6\end{aligned} $$ | |
| 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: $$ 1 \color{blue}{-4i} + \color{red}{4i^2} \color{blue}{-4i} + \color{green}{16i^2} \color{orange}{-16i^3} + \color{green}{4i^2} \color{orange}{-16i^3} +16i^4 = \\ = 16i^4 \color{orange}{-32i^3} + \color{green}{24i^2} \color{blue}{-8i} +1 $$Find $ \left(-1+2i\right)^3 $ in two steps. S1: Swap two terms inside bracket S2: apply formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 2i $ and $ B = 1 $. $$ \left(-1+2i\right)^3 \xlongequal{ S1 } \left(2i-1\right)^3 = \left( 2i \right)^3-3 \cdot \left( 2i \right)^2 \cdot 1 + 3 \cdot 2i \cdot 1^2-1^3 = 8i^3-12i^2+6i-1 $$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 }$. $$ \begin{aligned}\left(-1+2i\right)^2& \xlongequal{ S1 } \left(1-2i\right)^2 \xlongequal{ S2 } \color{blue}{1^2} -2 \cdot 1 \cdot 2i + \color{red}{\left( 2i \right)^2} = \\[1 em] & = 1-4i+4i^2\end{aligned} $$ |
| 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 | $$ 24i^2 = 24 \cdot (-1) = -24 $$$$ 8i^3 = 8 \cdot \color{blue}{i^2} \cdot i =
8 \cdot ( \color{blue}{-1}) \cdot i =
-8 \cdot \, i $$ |
| ⑧ | $$ -12i^2 = -12 \cdot (-1) = 12 $$$$ 4i^2 = 4 \cdot (-1) = -4 $$ |
| ⑨ | Combine like terms: $$ \color{blue}{16} + \color{red}{32i} \color{green}{-24} \color{red}{-8i} + \color{green}{1} = \color{red}{24i} \color{green}{-7} $$Combine like terms: $$ \color{blue}{-8i} + \color{red}{12} + \color{blue}{6i} \color{red}{-1} = \color{blue}{-2i} + \color{red}{11} $$Combine like terms: $$ \color{blue}{1} -4i \color{blue}{-4} = -4i \color{blue}{-3} $$ |
| ⑩ | Multiply $ \color{blue}{3} $ by $ \left( 24i-7\right) $ $$ \color{blue}{3} \cdot \left( 24i-7\right) = 72i-21 $$Multiply $ \color{blue}{12} $ by $ \left( -2i+11\right) $ $$ \color{blue}{12} \cdot \left( -2i+11\right) = -24i+132 $$Multiply $ \color{blue}{39} $ by $ \left( -4i-3\right) $ $$ \color{blue}{39} \cdot \left( -4i-3\right) = -156i-117 $$ |
| ⑪ | Combine like terms: $$ \color{blue}{72i} \color{red}{-21} \color{blue}{-24i} + \color{red}{132} = \color{blue}{48i} + \color{red}{111} $$ |
| ⑫ | Combine like terms: $$ \color{blue}{48i} + \color{red}{111} \color{blue}{-156i} \color{red}{-117} = \color{blue}{-108i} \color{red}{-6} $$ |
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