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: $$ \color{blue}{-6i} + \color{blue}{8i} \color{red}{-12} + \color{red}{1} = \color{blue}{2i} \color{red}{-11} $$ |
This page was created using
Complex Expression calculator