Answer
$$(1+i)^3=2i-2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1+3i+3i^2+i^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1+3i-3-i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2i-2\end{aligned} $$ | |
| 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 $. $$ \left(1+i\right)^3 = 1^3+3 \cdot 1^2 \cdot i + 3 \cdot 1 \cdot i^2+i^3 = 1+3i+3i^2+i^3 $$ |
| ② | $$ 3i^2 = 3 \cdot (-1) = -3 $$ |
| ③ | $$ i^3 = \color{blue}{i^2} \cdot i =
( \color{blue}{-1}) \cdot i =
- \, i $$ |
| ④ | Combine like terms: $$ \color{blue}{3i} \color{blue}{-i} \color{red}{-3} + \color{red}{1} = \color{blue}{2i} \color{red}{-2} $$ |
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