Question
Answer
$$(1+2i)^3=-2i-11$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+2i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1+6i+12i^2+8i^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1+6i-12-8i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-2i-11\end{aligned} $$ | |
| 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 $. $$ \left(1+2i\right)^3 = 1^3+3 \cdot 1^2 \cdot 2i + 3 \cdot 1 \cdot \left( 2i \right)^2+\left( 2i \right)^3 = 1+6i+12i^2+8i^3 $$ |
| ② | $$ 12i^2 = 12 \cdot (-1) = -12 $$ |
| ③ | $$ 8i^3 = 8 \cdot \color{blue}{i^2} \cdot i =
8 \cdot ( \color{blue}{-1}) \cdot i =
-8 \cdot \, i $$ |
| ④ | Combine like terms: $$ \color{blue}{6i} \color{blue}{-8i} \color{red}{-12} + \color{red}{1} = \color{blue}{-2i} \color{red}{-11} $$ |
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