Question
Answer
$$(1+i)^3+6(1+i)^2+18\cdot(1+i)=32i+16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+i)^3+6(1+i)^2+18\cdot(1+i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1+3i+3i^2+i^3+6(1+2i+i^2)+18\cdot(1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1+3i-3-i+6(1+2i-1)+18\cdot(1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2i-2+6\cdot2i+18\cdot(1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2i-2+12i+18+18i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}14i-2+18+18i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}32i+16\end{aligned} $$ | |
| ① | Find $ \left(1+i\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 1 $ and $ B = i $. $$ \left(1+i\right)^3 = 1^3+3 \cdot 1^2 \cdot i + 3 \cdot 1 \cdot i^2+i^3 = 1+3i+3i^2+i^3 $$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 }$. $$ \begin{aligned}\left(1+i\right)^2 = \color{blue}{1^2} +2 \cdot 1 \cdot i + \color{red}{i^2} = 1+2i+i^2\end{aligned} $$ |
| ② | $$ 3i^2 = 3 \cdot (-1) = -3 $$ |
| ③ | $$ i^3 = \color{blue}{i^2} \cdot i =
( \color{blue}{-1}) \cdot i =
- \, i $$$$ i^2 = -1 $$ |
| ④ | Combine like terms: $$ \color{blue}{1} + \color{red}{3i} \color{blue}{-3} \color{red}{-i} = \color{red}{2i} \color{blue}{-2} $$Combine like terms: $$ \, \color{blue}{ \cancel{1}} \,+2i \, \color{blue}{ -\cancel{1}} \, = 2i $$ |
| ⑤ | Multiply $ \color{blue}{18} $ by $ \left( 1+i\right) $ $$ \color{blue}{18} \cdot \left( 1+i\right) = 18+18i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{2i} -2+ \color{blue}{12i} = \color{blue}{14i} -2 $$ |
| ⑦ | Combine like terms: $$ \color{blue}{14i} \color{red}{-2} + \color{red}{18} + \color{blue}{18i} = \color{blue}{32i} + \color{red}{16} $$ |
This page was created using
Complex Expression calculator