Question
Answer
$$3(1+2i)^4+12(1+2i)^3+39(1+2i)^2=60i-270$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3(1+2i)^4+12(1+2i)^3+39(1+2i)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3(16i^4+32i^3+24i^2+8i+1)+12(1+6i+12i^2+8i^3)+39(1+4i+4i^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}3(16-32i-24+8i+1)+12(1+6i-12-8i)+39(1+4i-4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}3(-24i-7)+12(-2i-11)+39(4i-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}-72i-21-24i-132+156i-117 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}-96i-153+156i-117 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}60i-270\end{aligned} $$ | |
| ① | $$ (1+2i)^4 = (1+2i)^2 \cdot (1+2i)^2 $$ |
| ② | Find $ \left(1+2i\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}{ 2i }$. $$ \begin{aligned}\left(1+2i\right)^2 = \color{blue}{1^2} +2 \cdot 1 \cdot 2i + \color{red}{\left( 2i \right)^2} = 1+4i+4i^2\end{aligned} $$ |
| 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 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 1 $ and $ B = 2i $. $$ \left(1+2i\right)^3 = 1^3+3 \cdot 1^2 \cdot 2i + 3 \cdot 1 \cdot \left( 2i \right)^2+\left( 2i \right)^3 = 1+6i+12i^2+8i^3 $$Find $ \left(1+2i\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}{ 2i }$. $$ \begin{aligned}\left(1+2i\right)^2 = \color{blue}{1^2} +2 \cdot 1 \cdot 2i + \color{red}{\left( 2i \right)^2} = 1+4i+4i^2\end{aligned} $$ |
| 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 | $$ 24i^2 = 24 \cdot (-1) = -24 $$$$ 12i^2 = 12 \cdot (-1) = -12 $$ |
| ⑧ | $$ 8i^3 = 8 \cdot \color{blue}{i^2} \cdot i =
8 \cdot ( \color{blue}{-1}) \cdot i =
-8 \cdot \, i $$$$ 4i^2 = 4 \cdot (-1) = -4 $$ |
| 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: $$ \color{blue}{1} + \color{red}{6i} \color{blue}{-12} \color{red}{-8i} = \color{red}{-2i} \color{blue}{-11} $$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: $$ \color{blue}{-72i} \color{red}{-21} \color{blue}{-24i} \color{red}{-132} = \color{blue}{-96i} \color{red}{-153} $$ |
| ⑫ | Combine like terms: $$ \color{blue}{-96i} \color{red}{-153} + \color{blue}{156i} \color{red}{-117} = \color{blue}{60i} \color{red}{-270} $$ |
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